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.
In general, though, the answer to this question would depend on the specifics of the argument in question. Sometimes you might be able to salvage something; maybe there's some other setting where same methods work, or where some hypothesis analogous to the false one ends up holding, or something like that. But of course from a purely logical perspective, if I prove that P implies Q and P turns out to be false, I've learned nothing about Q.