Could you give an example of how this works?

I know a couple of US-born people who have successfully relinquished their US citizenship, and were not made to pay any money to the US as they were not renouncing. One of them says she does not get hassled when entering the US (due to ignorance of the rules) because her place of birth is a place which also exists outside the US, and it is just listed as that in her passport. The other was born in New York and has not been hassled either but he tries to avoid travel to the US.

I think that if you want to relinquish your US citizenship because you genuinely do not have ties to the US and would only visit for the same reason as a regular citizen of your new country, then you should do so without worrying about problems entering as a tourist.