This is one of those reasons that those looking into renouncing get counseled to consider all possible consequences carefully. The laws exist on the books to prohibit you from being admitted to the US, even as a "tourist" - though at present I haven't heard about them being enforced much (if at all). I know when I was looking into it, I decided I could not renounced until after the death of both my parents, simply on the off chance that I would "need" to go back in order to care for them or deal with their estates.
In theory it should just be a matter of traveling with your CLN (Certificate of Loss of Nationality) in the off chance that an Immigration officer notices your US birthplace on your non-US passport. But with the current ESTA system, it may well depend on them "codifying" that by somehow including something in the online system to recognize the fact of someone having renounced. Unfortunately, the CLNs don't seem to be subject to any sort of central data base and the format of the certificates handed out by the consulates varies greatly from one consulate to the next - so it's not like you can just pop in your CLN number so that the system can check it out.