There is a curious passage in Romans 5.7-8 that perhaps suggests incarnation:
Indeed, rarely will anyone die for a righteous person--though perhaps for a good person someone might actually are to die. But God proves his love for us in that while we were still sinners Christ died for us.
The major premise of the argument here is clear: to die for a person is a fantastic display of love. But if the minor premise is that Christ died for us, how does the conclusion follow that this is a fantastic display of God's love for us? The implication, to my mind, is that God is somehow so intimately connected with Christ that the latter's death counts as the former's.
Now the notion of God's death or God's dying has been a historically controversial one, as far as I know. Luther spoke of the death of God in the death of Christ, and we all know Moltmann's treatment of that issue in The Crucified God. The church fathers, on the other hand, if I am not mistaken, did not go so far as to affirm that God had died. They preferred to make the qualification that he had died in his human nature only.
If my proposed interpretation of the Pauline passage is correct, then there may be biblical support for using the language of God's dying. It would no doubt be true that he dies in his human nature, since only this is subject to mortality, but that doesn't invalidate the less specified statement "God died."
On the other hand, it would seem to me that the classical language describing Mary as theotokos, mother/bearer of God, presupposes a willingness to make unqualified statements about God which ground their truth in his human nature. Mary is not the mother of God in the sense of having produced a previously nonexistent deity; it is through Christ's human nature that the mother-of-God language can be used. Still, it is used without the qualifications "in his human nature," etc.
Along the same lines, then, perhaps we can use the language of the death of God for our sins without the qualifications and specifications of "in his humanity only," etc.