It depends. The most likely scenario would be that RH holds except in very specific conditions. Then, any dependent theorems would inherit the same conditions. In many cases, those conditions may not affect the dependent theorem, so they'd still be completely valid. In some cases, those conditions may make the dependent theorem useless, like if RH was "all numbers are even", and your theorem was "all numbers % 2 equal zero, because we know even numbers % 2 are zero and we assume RH", then the exception to RH "except odd numbers" would make your theorem devolve to "all numbers % 2 are zero except the odd ones, because we know even numbers % 2 are zero", which is obviously just a restatement of an existing statement.

In other cases, the new condition affects your theorem but doesn't completely invalidate it. So you can either accept that your theorem is weaker, or find other ways to strengthen it given the new condition.

That's all kind of abstract though. I'm not an expert on RH or what other important math depends on it holding up. That would be interesting to know.