What is filter memory?
Filter memory refers to the effect of past input on the current input. Infinite impulse response (IIR) filters and finite impulse response (FIR) filters are fundamentally different from the perspective of their memory. The memory of an FIR will essentially be the number of taps, while IIR has theoretically infinite memory (although when they are stable, the memory passes under the numerical accuracy of the system beyond a certain point).

More on this http://gregstanleyandassociates.com/whitepapers/FaultDiagnosis/Filtering/Filter-Memory/filter-memory.htm.