Changing film capacitors to ceramic capacitors in pulsed induction metal detector
I am aware that the capacitance of ceramic capacitors changes with the
DC offset.

You can replace them all with ceramics but use X7R as a default choice because these don't drift much with applied voltage. Here you can see that X7R is the best in those graphed and the bigger body the have, the more stable they are so choose 1210 or 1812: -

Bottom line is RTFD (read the fine detail is one version of this acronym). So read the fine data sheet also.
There are several marked as 5% tolerance, the highest value being 22 nF so use C0G/NP0 dielectric to get decent stability over and above those shown in the graph. They all appear to be used in timing circuits and you don't want those values to change much.

Are there any film capacitors in the circuit that should definitely
NOT be replaced with ceramic capacitors? If so, why?

No, they can all be replaced with ceramic capacitors but choose the correct dielectric as outlined above.