If a premise is false, can it still support the conclusion?

Depends on your subject if you are just talking about philosophy - logic yes, you assume that your premise is correct. Even if it is total bogus. Premises can be incorrect and still be a valid argument. For example:
If it is wet, then it is raining
It is wet
Theirfore it is raining.
Their are other explinations for why it might be wet but you just go with what your premise tells you.