It used to be that you have to pay a big fat one-time fee to renounce, as kind of a substitute for all of your future tax payments.

I think they changed the rules on that recently, though.

Edit: since GP made the same point about the exit tax, if you purport to renounce without paying the fee, then you haven’t really renounced. You and your assets can be seized for tax evasion (assuming you stop filing tax returns) if they ever touch US jurisdiction again, or any country that has a tax treaty with US, or countries with extradition to US.