Answer
All POI files are stored in the same folder as the map they belong to.
E.g.: a POI file containing all the cheap gas stations in Western Europe can be found in the folder that contains the map of Western Europe (usually called ‘Western Europe’).
If you are using a TomTom RIDER, GO or ONE, map folders can be found on the storage card (or on the internal card) in your TomTom (e.g. ‘F:\\’).
If you are using a PDA with TomTom Navigator software, map folders can be found on the storage card in your PDA (‘\\Storage Card\\’).
For newer TomTom devices (since around 2016), it is only possible to access POI files on the device itself or via Tyre/MyDrive.