I would add from James 1:
12 Blessed [is] the man that endureth temptation: for when he is tried, he shall receive the crown of life, which the Lord hath promised to them that love him. 13 Let no man say when he is tempted, I am tempted of God: for God cannot be tempted with evil, neither tempteth he any man: 14 But every man is tempted, when he is drawn away of his own lust, and enticed. 15 Then when lust hath conceived, it bringeth forth sin: and sin, when it is finished, bringeth forth death. 16 Do not err, my beloved brethren. 17 Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning.
and, preceding those quoted by sscott
Heb 2:16 For verily he took not on [him the nature of] angels; but he took on [him] the seed of Abraham.
Romans 8:3 For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh:
It is important to separate 'sin' (the cause of sins) and 'sinful flesh'. People do not sin only because of sinful flesh. Sinful flesh is fallen flesh. Originally, Adam's flesh was not fallen, and he had a spirit which was free from sin which could fellowship with God - according to Luke, his Father - Luke 3:38.
So, what did unfallen flesh look like, if the spirit inside was free from all sin? That's an unanswerable question, but, some preachers would suggest that there was a glory gilding the first man and woman which was forced to depart because of sin. Christ came without the 'glory', also. Maybe read Romans 5 as well.
I'm suggesting that while Christ had no sin within Him, and, He did not DESIRE to sin (because He was the Son of God), He did have to contend with the urges and temptations to His flesh, because that's what Paul seems to be implying in Rom 8:3. But the word 'likeness' is extremely important.
Psalm 22 is also worth reading, as there is a breakthrough point which is repeated in Heb 2:12. The first section, from v 1 - v 21, ably describe what Christ was experiencing on the cross. There is a lot of meaning hidden in those verses: how he was reminding himself of truth which he had known, and, how the various spiritual pressures were manifesting themselves to him, which he had to overcome without sinning, to destroy the devil (on the cross - Heb 2:14 Forasmuch then as the children are partakers of flesh and blood, he also himself likewise took part of the same; that through death he might destroy him that had the power of death, that is, the devil; 15 And deliver them who through fear of death were all their lifetime subject to bondage.)
If Christ had ever sinned, for whatever reason, He would not have been able to give His life in substitutionary sacrifice for mankind. He chose to not sin.
Through new birth, we also are brought into a spiritual condition which enables us to choose to 'not sin': 1 John 3:9 'and he can not sin'.
This could be taken as a reverse proof that Christ did not sin, because, we are made like Him: 1 John 3:1 Behold, what manner of love the Father hath bestowed upon us, that we should be called the sons of God: therefore the world knoweth us not, because it knew him not. 2 Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is. 3 And every man that hath this hope in him purifieth himself, even as he is pure.
