But, in addition to the passages I gave, we also have to consider Romans 5:12, 18-19 and 8:3:
Rom 5:12 Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men because all sinned—
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Rom 5:18 Therefore, as one trespass led to condemnation for all men, so one act of righteousness leads to justification and life for all men.
Rom 5:19 For as by the one man's disobedience the many were made sinners, so by the one man's obedience the many will be made righteous. (ESV)
Rom 8:3 For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh, (ESV)
Note that in Romans 5, sin is inherited from Adam's "one trespass." That one trespass is what then makes our flesh sinful, as Rom 8:3 states. However, Paul doesn't say that Jesus came "in sinful flesh," but in the "likeness of sinful flesh." What we can see then is that Jesus was fully human (John 1:14; Phil 2:7-8; 1 John 4:2), but that he was without the original taint of sin that we all have.
Jesus is the second Adam, who, like how the first Adam was initially, was without sin in the flesh. But, unlike the first Adam who sinned and brought the curse on all flesh, Jesus remained sinless. I believe that is the distinction Paul is making with the use of "likeness"--Jesus is truly human, yet he didn't have flesh tainted by sin as the rest of us do. He needed to use "likeness" because he also used "sinful," unlike John who simply says, "became flesh" (John 1:14) and "in the flesh" (1 John 4:2). How else could Paul say that Jesus was truly human, yet was not sinful, by participating in "sinful flesh"?
1Pe 2:11 Beloved, I urge you as sojourners and exiles to abstain from the passions of the flesh, which wage war against your soul. (ESV)
That seems to me to be the best understanding of the entire matter, and that Jesus was tempted, but his temptations came from the devil, from without, not from within as we experience with "passions of the flesh." I think that the idea that humans are born with a sinful (or fallen) nature is without question the biblical witness. If we weren't, it would be possible that someone could live perfectly and not need Christ's atoning sacrifice.