Can a child born in US to Indian Father and Pakistani Mother obtain an Indian Citizenship or OCI card?

Would it be possible for him to obtain Indian Citizenship for him? (I assume not since he will automatically have US citizenship)

No.

I want to note that it's not true that he can't have Indian citizenship just because he will automatically have US citizenship. According to India's citizenship act, a child born abroad to at least one Indian citizen parent will be an Indian citizen by descent if he/she is registered at an Indian consulate within one year of birth with the declaration that he/she does not possess a foreign passport. Your son having US citizenship doesn't the ability to meet that requirement, as long as you don't get him a US passport.

However, since both of you are US citizens, neither of you are Indian citizens when your son is born (you lost your Indian citizenship when you voluntarily acquired US citizenship), and thus neither of you have Indian citizenship that can be passed to him.

Alternately is it possible for him to obtain an OCI card? (I've heard that if one's parents or grandparents are from Pakistan or Bangladesh, then he/she is not eligible to apply for an OCI card.

I am not very familiar with OCI, but I believe what you heard is correct -- your son can't get OCI because one of his parents is or was a Pakistani citizen.