Does original sin have an impact on Jesus?

According to Hebrews 4:15 Jesus was without sin.

Hebrews 4:15 For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin. (KJV)

Your question appears to be in two parts:

1) Was Jesus subject to original sin, and 2) Was Jesus born without a sin nature

and the short answer is no and yes, respectively. A fuller answer will reveal that the no to question 1 is because of the yes of question 2.

adjective adjective: subject /ˈsəbjekt/

1.
likely or prone to be affected by (a particular condition or occurrence, typically an unwelcome or unpleasant one).
"he was subject to bouts of manic depression"

In Genesis chapter 2 we see God placing the man He had created in a garden to "work it and keep it". We also see God commanding lavish provision for Adam along with one prohibition. To this prohibition was attached a stated consequence, namely death. All of this took place before Eve was created. There is no record in Scripture of God reiterating his prohibition personally to Eve so, whatever Eve knew of God's command (and it is clear she knew) she likely knew secondhand from Adam. This is well in keeping with Adam's commission which was to work (or tend) the garden and to keep (or protect it). It is also another aspect of his failure as he should have protected her from breaching the prohibition.

So when the Serpent comes on the scene in Genesis chapter 3 and begins his temptation he subtly goes to the one with the less direct knowledge and we have it that

Adam was formed first, then Eve; and Adam was not deceived, but the woman was deceived and became a transgressor. - 1 Timothy 2:13-14

Then, because Adam's transgression was not one of falling to deception (like Eve) but one of direct, first order disobedience we have it that

sin came into the world through one man, and death through sin -Romans 5:12

many died through one man's trespass - Romans 5:15

the judgment following one trespass brought condemnation - Romans 5:16

because of one man's trespass, death reigned through that one man - Romans 5:17

by the one man's disobedience the many were made sinners - Romans 5:19

It is in this way that what some theologians have coined 'original sin' has passed to all humanity; through Adam. In order to understand what original sin is it is necessary to examine what the temptation was since God did not describe any aspect of the forbidden fruit other than the name of the tree that bore it (the knowledge of good and evil) and the consequence of eating it (death). While it is true that on the most superficial level "God said 'no' and Adam did 'yes'" is an accurate rendering of the disaster it does little to help us understand what has been passed on to all humanity through Adam.

For God knows that when you eat of it your eyes will be opened, and you will be like God, knowing good and evil. - Genesis 3:5

Here is the heart of the temptation: God is trying to keep you from becoming like him in this specific area of possessing knowledge of good and evil. We see that confirmed in two places:

So when the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was to be desired to make one wise,[b] she took of its fruit and ate, and she also gave some to her husband who was with her, and he ate - Genesis 3:6

Then the Lord God said, “Behold, the man has become like one of us in knowing good and evil. - Genesis 3:22

We see from verse 6 that Adam was present with Eve while the serpent deceived her and he too ate of the fruit, essentially rejecting God as rightful definer of good and evil and claiming that right for himself thereby making himself like God. It was a coup of the highest magnitude.

Since there is an order in creation whereby like begets like and since every person is descended from Adam (even Eve who was created from his flesh) therefore every person inherits this rebellious disposition and is subject to it; "I, not God, will determine what is good and evil" and by this disposition everyone dies.

Oswald Chambers has nicely summed up this concept in his devotional work My Utmost For His Highest:

Wherefore as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned. — Romans 5:12

The Bible does not say that God punished the human race for one man’s sin; but that the disposition of sin, viz., my claim to my right to myself, entered into the human race by one man, and that another Man took on Him the sin of the human race and put it away (Hebrews 9:26) — an infinitely profounder revelation. The disposition of sin is not immorality and wrong-doing, but the disposition of self-realization — I am my own god. This disposition may work out in decorous morality or in indecorous immorality, but it has the one basis, my claim to my right to myself. When Our Lord faced men with all the forces of evil in them, and men who were clean living and moral and up right, He did not pay any attention to the moral degradation of the one or to the moral attainment of the other; He looked at something we do not see, viz., the disposition.

Sin is a thing I am born with and I cannot touch it; God touches sin in Redemption. In the Cross of Jesus Christ God redeemed the whole human race from the possibility of damnation through the heredity of sin. God nowhere holds a man responsible for having the heredity of sin. The condemnation is not that I am born with a heredity of sin, but if when I realize Jesus Christ came to deliver me from it, I refuse to let Him do so, from that moment I begin to get the seal of damnation. “And this is the judgment” (the critical moment) “that the light is come into the world, and men loved the darkness rather than the light.”

Because God does not change nor go back on his gifts or callings and since it was to the man the prohibition was uttered and through the man that sin and death came into the world so it is that through the man sin is passed down through the process of procreation and through a Man salvation must come.

The virgin birth of Christ is critically important then, not necessarily because Mary was chaste (although she certainly was) but because no human male was involved in the conception. In this way Mary, though having inherited a sin nature of her own because of her descent from Adam, was able to bear a child who inherited no 'original sin' since sin does not pass through woman but through man.

So, in answer to your question, Jesus was not the subject (under the influence) of original sin precisely because he was born without a sin nature. He was the second Adam and, just like the first Adam, had no sin via his creation. However, unlike the first Adam, Jesus did not subsequently acquire sin through volition and so he became the author of eternal salvation. and the first born from among the dead.

Does original sin have an impact on Jesus?

Like any coin worth it weight in gold, this question has two sides of an equation, just like a coin of great value.

The question is both yes and no!

Let me try to explain this.

On one side of the coin, Jesus was conceived without original sin and as such was spared from all stain of sin inherited from Adam’s disobedience.

It is true that some denominations believe that Mary was conceived without original sin by a unique privilege bestowed onto her by Almighty God. Catholics call this the Immaculate Conception of the Blessed Virgin Mary.

Furthermore, St. Thomas Aquinas maintains that it was indeed possible for Our Lord Jesus Christ to have been born of sinful woman; Christ's glory would not have been diminished thereby. But, he states elsewhere, it would have been inappropriate:

God so prepares and endows those, whom He chooses for some particular office, that they are rendered capable of fulfilling it, according to 2 Cor. 3:6: "(Who) hath made us fit ministers of the New Testament." Now the Blessed Virgin was chosen by God to be His Mother. Therefore there can be no doubt that God, by His grace, made her worthy of that office. ... But she would not have been worthy to be the Mother of God, if she had ever sinned. First, because the honor of the parents reflects on the child, according to Prov. 17:6: "The glory of children are their fathers": and consequently, on the other hand, the Mother's shame would have reflected on her Son. Secondly, because of the singular affinity between her and Christ, who took flesh from her: and it is written (2 Cor. 6:15): "What concord hath Christ with Belial?" Thirdly, because of the singular manner in which the Son of God, who is the "Divine Wisdom" (1 Cor. 1:24) dwelt in her, not only in her soul but in her womb. And it is written (Wis. 1:4): "Wisdom will not enter into a malicious soul, nor dwell in a body subject to sins."

We must therefore confess simply that the Blessed Virgin committed no actual sin, neither mortal nor venial; so that what is written (Cant 4:7) is fulfilled: "Thou art all fair, O my love, and there is not a spot in thee," etc. - Was the dogma of the Immaculate Conception formulated in response to someone challenging Mary's ability to bear a perfect Son?

Also according to St. Thomas Aquinas only Adam was responsible for transmitting original sin to the human race and thus it follows that only the Holy Spirit was responsible for preserving Christ from original sin, not Mary.

St. Thomas Aquinas in his Summa clearly states that it was Adam and not Eve that was responsible for the transition of original sin. Now if Mary, the Mother of Jesus is the New Eve and Jesus Christ is the New Adam, it is obvious that the Immaculate Conception was not necessary for our salvation for the Holy Spirit can operate in that dwelling place He so chooses to. It the Holy Spirit that made it possible for Christ to be free of original sin and not Mary, the New Eve.

The solution of this question is made clear by what has been said. For it has been stated that original sin is transmitted by the first parent in so far as he is the mover in the begetting of his children: wherefore it has been said that if anyone were begotten materially only, of human flesh, they would not contract original sin. Now it is evident that in the opinion of philosophers, the active principle of generation is from the father, while the mother provides the matter. Therefore original sin, is contracted, not from the mother, but from the father: so that, accordingly, if Eve, and not Adam, had sinned, their children would not contract original sin: whereas, if Adam, and not Eve, had sinned, they would contract it. - Whether if Eve, and not Adam, had sinned, their children would have contracted original sin?

Just as St. Paul states, Jesus is without sin.

For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin. (KJV) - Hebrews 4:15

Now on the other side of the coin, all mankind was stained with both original sin and actual sin and this did have an impact Jesus because he died saving sinful mankind from their sins.

Jesus Our High Priest was born in order to save man from their sins. Jesus suffered his Passion and Death for our salvation. Only in this sense did original sin have any impact on Jesus who is called the Christ.

Although he himself had never been stained with original sin; he did suffer his Passion and Death for mankind who truly did suffer for original and actual sin.