FIR Filter on Cortex M0
Of course you can. You can implement any kind of DSP on a Cortex M0. Practically it will depend on the size of your filter (ie. RAM usage) and required speed (if it is real-time). Of course the implementation will have to be fixed-point, as M0 does not have a floating-point unit (so floating point operations are slow).

I recommend using CMSIS DSP which is a DSP library provided by ARM and optimized specially for their cores.