Do parts of Hebrews 3 and 4 imply that saved Christians can become unsaved?

In Hebrews 3:11-13, Paul uses the example of the Hebrew exodus to bring home a lesson to Christians. In that instance, the people of God were being brought out of slavery into a land of milk and honey. The problem then, as now, is that the people were complaining about their discomforts all the way there rather than praising God for where they were going and enduring their trials with fortitude.

The Promised Land is not symbolic of Heaven. It is symbolic of freedom from the slavery of Egypt, freedom to love God with all the heart, soul, mind, and strength. The lesson is that Christians may lose sight of where God is leading them. It's an item of faith in a God who is able, because like the Israelites trudging through the wilderness, we also go through dry times in this world during our traveling/training period. The promise of God is love, joy, peace and the other fruits of the Spirit -- and few church-goers make it there.

The danger is that we never enter into the life where one is fully led by the Spirit to conquer the flesh, as Joshua was led and conquered the enemies in the land. (We will not wage warfare in Heaven!) Instead of us being pushed here and there by all our sins, frailties and foibles, God's promise is that we would be at "rest" and abiding in Him. Without faith in God during this lifetime-long journey, we will never make it.

This has nothing to do with salvation in the sense of receiving Jesus and then losing the possibility of Heaven later. It has everything to do with never letting God's Spirit achieve all He can do in a life submitted to Him.

To add to Thomas' answer, you can also refer to the parable of the sower, Mark 4:3-20, especially verses 16 and 17. Here, Jesus talks about people that are also described there in Hebrews 4:2, who were excited when they first heard the message but who did not (to continue the sower's metaphor) allow it to take root. They weren't transformed by the message, they were only excited about for some shallow, selfish reason (perhaps they believed God would make them wealthy or cure their illness and then leave them free to pursue their own desires again).

Yes.

And that's consistent with the rest of New Testament teaching.

Jesus himself taught that some would abandon their faith:

Matthew 24:10-13 (NRSV)

Then many will fall away, and they will betray one another and hate one another. And many false prophets will arise and lead many astray. And because of the increase of lawlessness, the love of many will grow cold. But the one who endures to the end will be saved. [emphasis mine]

Note the future tense. This is because, even though Jesus said, "It is finished," as he died on the cross, there is another sense in which salvation is a lifelong process.

Paul dealt with the future aspect of salvation in his letter to the Philippians, stressing God's ongoing work to complete the process:

Philippians 2:12-13 (NRSV)

[W]ork out your own salvation with fear and trembling; for it is God who is at work in you, enabling you both to will and to work for his good pleasure.

Paul expressed confidence that the readers of this letter would endure to the end:

Philippians 1:6-7 (NRSV)

I am confident of this, that the one who began a good work among you will bring it to completion by the day of Jesus Christ. It is right for me to think this way about all of you, because you hold me in your heart, for all of you share in God's grace with me, both in my imprisonment and in the defense and confirmation of the gospel.

However, in his own life he recognized there was a lot of work left to do:

Philippians 3:10-12 (NRSV)

I want to know Christ and the power of his resurrection and the sharing of his sufferings by becoming like him in his death, if somehow I may attain the resurrection from the dead. Not that I have already obtained this or have already reached the goal; but I press on to make it my own, because Christ Jesus has made me his own.

And in 1 Corinthians he expressed his concern about not being able to endure:

1 Corinthians 9:25-27 (NRSV)

Athletes exercise self-control in all things; they do it to receive a perishable wreath, but we an imperishable one. So I do not run aimlessly, nor do I box as though beating the air; but I punish my body and enslave it, so that after proclaiming to others I myself should not be disqualified.

Back to Hebrews. Not only does Hebrews say once-saved Christians can fall away, it warns that the consequences for this deliberate rejection of Christ are severe:

Hebrews 6:4-6 (NRSV)

For it is impossible to restore again to repentance those who have once been enlightened, and have tasted the heavenly gift, and have shared in the Holy Spirit, and have tasted the goodness of the word of God and the powers of the age to come, and then have fallen away, since on their own they are crucifying again the Son of God and are holding him up to contempt.

_{[from the perspective of a Wesleyan Arminian.]}

No.

My sheep hear my voice, and I know them, and they follow me. ²⁸I give them eternal life, and they will never perish, and no one will snatch them out of my hand. ²⁹My Father, who has given them to me, is greater than all, and no one is able to snatch them out of the Father's hand. - John 10:27-29 ESV

and

I will not leave you or forsake you. - Josh. 1:5 ESV

and

For the LORD will not forsake his people; - Ps. 94:14 ESV

The logical explanation of the verses in Hebrews is that those who fall away were never truly participators in the covenant to begin with. Indeed, reading the surrounding context in Hebrews is quite helpful:

So we see that they were unable to enter because of unbelief. Therefore, while the promise of entering his rest still stands, let us fear lest any of you should seem to have failed to reach it. ²For good news came to us just as to them, but the message they heard did not benefit them, because they were not united by faith with those who listened. ³For we who have believed enter that rest... - Hebrews 3:19-4:3 ESV

As the author explains, those who did not enter never believed in the first place. This may seem strange considering what appeared to be a strong faith, so it's helpful to go to James on the subject.

What good is it, my brothers, if someone says he has faith but does not have works? Can that faith save him? ...You believe that God is one; you do well. Even the demons believe - and shudder! - James 2:14, 19 ESV

Typically faith is split into three categories: knowledge, assent, commitment. Most modern day Americans have knowledge (that is, most know of the gospel of Christ, or at least have heard of Jesus). Many believe that He died for their sins. But as James says, "even the demons believe!" Not so many, perhaps, actually trust God for their salvation. It is one thing to think that Jesus died for you, pray a prayer, and move on. It is another to live one's whole life utterly devoted to God, in light of Christ's death and wholly dependent upon that death for Salvation.

So, it seems the people in the church addressed in Hebrews were knowers, believers even. But "faith by itself, if it does not have works, is dead." (James 4:17). And dead faith is really no use to anyone. With a dead (nonexistent, i'd argue) faith, those people never really trusted in God in the first place - they were never Jesus's sheep in the first place, so they were not snatched out of his hand.

_{[from the perspective of a reformed presbyterian.]}