Is belief in the suspension or replacement of any of the previous covenants necessary for salvation today?

No, what people believe about the Old Testament Covenants is not essential to salvation, although a proper understanding of why God introduced those Covenants is critical to appreciating the significance of the New Covenant. All of the Old Testament Covenants point forward to the coming of Christ Jesus who introduced the New Covenant the night he was betrayed. That Covenant is not limited to Jews, but is open to any and all who acknowledge they are sinners in need of a saviour, and who repent and turn to God.

Before Jesus came to earth the basis for salvation was faith in God and in God’s promises. Hebrews chapter 11 lists many Old Testament characters who, because of their faith in God, were assured of being the recipients of God’s promises:

”All these people were still living by faith when they died. They did not receive the things promised; they only saw them and welcomed them from a distance. And they admitted that they were aliens and strangers on earth... Instead, they were longing for a better country – a heavenly one. Therefore God is not ashamed to be called their God, for he has prepared a city for them” (Hebrews 11:13-16).

Today, faith is still essential to salvation. Here are a few relevant Bible verses:

”But God, being rich in mercy, because of the great love with which he loved us, even when we were dead in our trespasses, made us alive together with Christ—by grace you have been saved— and raised us up with him and seated us with him in the heavenly places in Christ Jesus, so that in the coming ages he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus. For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, not a result of works, so that no one may boast. For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them” (Ephesians 2:4–10).

The means God has chosen to bestow His grace upon us is through faith. “Now faith is being sure of what we hope for and certain of what we do not see” (Hebrews 11:1). Salvation is obtained by faith in God’s Son, Jesus Christ, in what He has done—specifically, His death on the cross and His resurrection. But even faith is not something we generate on our own. Faith, as well as grace, is the gift of God (Ephesians 2:8). He bestows saving faith and saving grace upon us in order to redeem us from sin and deliver us from its consequences. So God saves us by His grace through the faith He gives us. Both grace and faith are gifts.

The book of Romans describes God’s plan of salvation. It shows how we must first acknowledge that we are sinners and place our trust in what Jesus Christ has done to atone for our sin. Salvation, the forgiveness of sins, is available to anyone who will trust in Jesus Christ as their Lord and Saviour.

“...if you confess with your mouth Jesus as Lord, and believe in your heart that God raised Him from the dead, you will be saved" (Romans 10:9).

"...for everyone who calls on the name of the Lord will be saved" (Romans 10:13).

Salvation depends on coming to saving faith in Jesus Christ and accepting that only through his death and resurrection can we receive forgiveness for our sins and be adopted into God’s family. As far as I know, all Christians understand this basic principle of salvation – faith in God and in what God has done, in Jesus Christ, to atone for our sins so that we may receive eternal life.

I found an article that explains how people were saved prior to the coming of Jesus and the introduction of the New Covenant. It shows how the Old Testament saints were aware of the promised Redeemer, and they were saved by faith in that Saviour, the same way people are saved today. There is no other way. Jesus is “‘the stone you builders rejected, which has become the capstone.’ Salvation is found in no one else, for there is no other name under heaven given to men by which we must be saved” (Acts 4:11-12, quoting Psalm 118:22). https://www.gotquestions.org/Old-Testament-salvation.html

Today some groups in Christianity use Bible verses that imply the Jews were abandoned by God after Jesus was crucified, and that Jesus brought others into a new covenant, which supposedly did away with the old covenant made with the nation of Israel at Mount Sinai. Those who maintain this promote what is called "Replacement Theology". They do not claim that one must believe that theology in order to be saved, though it does impact on their attitudes to the nation of Israel, and, if it is wrong, would indicate they have a distorted view of the biblical gospel of Jesus Christ. That is why it is an important question to resolve.

Replacement theologists might, for example, quote Jesus' words in Matthew 23:37-39 about Jerusalem not being willing to be gathered to Christ, so Jesus says, "Look, your house is abandoned to you", followed by his prophecy about the city and its temple being destroyed. That happened in A.D. 70. Some groups then form the conclusion that God abandoned the nation utterly (as a nation, even though individual Jews became Christians, thus entering into the new covenant Jesus inaugurated with his shed blood).

Well, history shows that God abandoned his people (for a season) on several occasions in the past, but that did not relinquish his everlasting covenant with the nation. Even though God sometimes punishes his people when they break their terms of that covenant, he never breaks his terms. Time and again, the Bible shows that God has made an everlasting covenant with the nation of Israel. He chose them even when they were a raggle-taggle bunch of nobodies in Egypt, brought them out and made them a nation to bear his Holy Name. Even when they were restored back to God's favour after he had used pagan nations to humiliate them, it was not because they deserved restoration, but because God's Name was to be honoured.

In Ezekiel 36 God spoke of a future time when he would sprinkle clean water on them, to cleanse them from the filth of their idolatry. "And I will give you a new heart, and I will put a new spirit in you. I will take out your stony, stubborn heart and give you a tender, responsive heart. And I will put my Spirit in you so that you will follow my decrees and be careful to obey my regulations... And when I bring you back, people will say, 'This former wasteland is now like the Garden of Eden... and everyone will know that I am the Lord." This despite the Edomites mistakenly assuming that God's judgment of his people and his abandonment of the Temple meant that his covenant with Israel was no longer in effect. The Edomites had boasted and elevated themselves against Israel, thinking they would replace Israel. Their destruction was thus guaranteed, which should serve as a warning to modern-day 'Edomites' - those who hate Israel and gloat over her difficulties.

Chapter 37 goes on to promise, "I will open your graves of exile and cause you to rise again. Then I will bring you back to the land of Israel... My servant David will be their king and they will have only one shepherd... And my servant David will be their prince forever. And I will make a covenant of peace with them, an EVERLASTING covenant... And when my Temple is among them forever, the nations will know that I am the Lord, who makes Israel holy." Notice that this promise was made long after King David had died. This is a future prophecy.

Today we see some Christians who say Israel has been cast off as God's chosen people and that now it is themselves who are in the new covenant, replacing Israel. Such arrogance! The majority view in Christianity holds that Jesus said Israel is a little flock, he is their one shepherd and he has other sheep not of that fold who he will also gather and make into one flock, with himself as their one shepherd. Revelation depicts 144,000 Israelites who get to heaven and sing a new song that nobody but they can master. There is also a great crowd of Gentiles in Heaven, praising God and Christ for their salvation.

Also, Romans chapters 9-11 states clearly. "I ask, then, has God rejected his own people, the nation of Israel? Of course not! I myself am an Israelite, a descendant of Abraham and a member of the tribe of Benjamin. No, God has not rejected his own people, whom he chose from the very beginning... I want you to understand this mystery, dear [Christian] brothers and sisters, so that you will not feel proud about yourselves. Some of the people of Israel have hard hearts, but this will last only until the full number of Gentiles comes to Christ. And so all Israel will be saved, as the scriptures say..."

Nowhere in the Bible does God say his covenant with them has ended. On the contrary, Galatians 3:16 shows they will yet acclaim the king foretold in Ezekiel.

I detail some of the reasons the Replacement theologians offer along with the reasons for disagreeing with them, in order to show that there are some Christians today who think God has replaced natural Israel with 'spiritual' Israel (i.e. Christians) but that many others would say God's everlasting covenant with natural Israel cannot be 'spiritualised away' to mean it no longer applies to that nation. However, there does not seem to be any group that insists you cannot be saved if you disagree with Replacement Theology. Christianity has always maintained that a person who exercises saving faith in Christ as the risen Son of God becomes a Christian and that further teaching (discipling) is needed to grasp the many deeper things of the faith.

Those who had been brought up under the old covenant, as a way of life, and who - subsequently - heard the gospel (which expresses the New Testament) were specifically warned not to turn back to the old covenant and were clearly warned of the dire consequences of doing so, by the writer of the epistle to the Hebrews.

We ought to give more earnest heed to the things which we have heard (Heb 2:1)

(says he - for he, also, heard these things from others and was not one of the original twelve who were with Jesus)

lest at any time we should let them slip.

He also says :

Take heed, brethren, (Heb 3:13)

(counting himself one with them)

lest there be in any of you an evil heart of unbelief in departing from the living God.

This departure is not to the ways of gentiles, to unmentionable sins and worldy luxuries, for what he is warning about is unbelief in the New Testament gospel. This departure from the living God is a potential departure from the new covenant back to an arrangement which merely foreshadowed the New Testament. But was not the reality.

Let us therefore fear, lest a promise being left of entering into rest, any of you should seem to come short of it

he says further, (Heb 4:1)

For unto us was the gospel preached as well as unto them.

The gospel was preached to the children of Israel as they engaged in an arrangement with God to display the coming New Testament in pictorial form - in artefact, in ritual, in sacrifice and in ordinance. But all of that was to be seen by faith. Not just to be ritually enacted, but to be understood. To be believed.

But the word preached .. (Heb 4:2)

(Note, 'preached'. The preaching was through demonstration. The word was expressed.)

did not profit them ... not being mixed with faith in them that heard.

They didn't actually believe it. They helped to enact it under an agreement with God (a covenant) but they never perceived the reality of it and never entered into it spiritually.

They needed to perceive that this - old - covenant was to be superseded. After which it would pass away and be but a memory. It was a picture. It was a demonstration. But what was to be believed was the reality.

Moreover that first covenant contained the law. It had to contain the law for mankind had put himself under law in Adam, choosing the tree of the knowledge of good and evil as a way of existing, favouring human understanding and human effort. (Rather than partaking of Life through receiving God's word.)

Thus the law was delivered - as part of the first covenant - at Sinai.

This do, and thou shalt live. (Luke 10:28)

But none ever 'did' the law. And none ever 'lived' through its ministration. It was a way of death, as forewarned in Eden.

In the day that thou eatest thereof ... thou shalt surely die. (Genesis 2:17)

So Paul also warns those who had heard the gospel from his lips, that after his departure they must, at all costs, reject those who would later come along and try to persuade them to mix old and new covenant. In this particular instance, the Galatian incident, it was an obligation to add circumcision to the gospel. But it can be any of a variety of ways of mixing old and new.

Whichever method it is, is irrelevant for Jesus had already made it very clear that

new cloth unto an old garment .. the rent is made worse (Matthew 9:16)

and -

new wine into old bottles ... the wine runneth out and the bottles perish.

One must - one must in the very nature of the covenants involved - thoroughly progress from the old to the new and, thereafter, fully appreciate that :

The law was a schoolmaster to bring to Christ. But after faith is come, we are no longer under a schoolmaster.(Galatians 3:24,25)

And Paul further warns them (of circumcision in this particular but it relates to any kind of turning back to the old covenant) -

I, Paul, say unto you that if ye be circumcised - Christ shall profit you nothing.

So, in summary, and in specific answer to the OP :-

To cling to the old covenant ... is to lose Christ.

And he, alone, is salvation.

Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved. (Acts 4:12)