No. 'Not eating or drinking' is the same as saying 'not eating and not drinking'. Consider it this way:
Not [eating or drinking]
[Not eating] or [not drinking]
If a person is '[not eating] or [not drinking]', then he is either not eating or he is not drinking.
EDIT: To clairfy, what we have here is a parallel construction. From The Cambridge Guide to English Usage:
The use of nor is probably declining, even in its core domain of coordinating two negative phrases.
- The gallery will not be open on Sundays or public holidays.
- The gallery will not be open on Sundays nor public holidays.
Both sentences are perfectly acceptable English, but the first shows that nor is not really needed to extend the negation over to 'public holidays.' Rather it may seem to overdo the expression of the negative for the purposes of a simple announcement. This use of nor for the second coordinate underscores the parallelism of the two phrases, and in the context of fine writing, with more extended coordinates, it would have its place.