You don't need to show your birth certificate for your son's admission. And if at all required, class 10th marksheet is a good enough proof.

For getting admission in medical College your son needs to carry the following documents-
1) neet admit card
2) neet rank letter
3) date of birth proof
4) class 10th pass certificate
5) class 12th pass certificate
6) class 12 the marksheet
7) 8 passport size photos,it should be same as the one used in application
8) provisional allotment letter
9) proof of identity
10) reserved candidate should carry proof of the same, issued by the authority