I have dual Bolivian/Czech Republic nationalities. In my Bolivian passport, I have a US B1/B2 visa, which will expire in three months. Can I apply for an ESTA and enter the US under the Visa Waiver Program (VWP) on my Czech passport, before my visa expires? The Czech passport is a new one with the chip.
If your current visa is of a category other than B-1/B-2, and therefore (at least in principle) is not valid for pure business/tourism trips, then it stands to reason that you can still apply for authorization for such a trip, which can either be a new visa or an ESTA for your Czech passport. There seems to be nothing that would prevent you from choosing the cheaper and simpler ESTA/VWP option.
If your current visa is a B-1/B-2 visa, then I would expect you can still apply for an ESTA for your Czech passport -- there's no rule saying that a dual national needs to use the same nationality's documentation for each of their trips. However, unless there's something concrete that prevents you from using the Bolivian passport, there would be little reason for you to want that.
Remember that the US rule is that the visa just has to be valid on the day you enter, and it is completely allowed for it to expire while you're still in the US.
Also, entering on an actual visa will put you in a slightly better legal position than entering under the VWP; for example you retain the possibility of applying for an extension or adjustment of status while in the US.
As far as I can tell from the answer CBP gave me on a similar question, the two passports of a dual citizen in the eyes of the CBP are completely independent. So my answer would be yes but note I am not a CBP officer, do not even play one on television and trying to apply logic to what the CBP does and says which might not be wise.