The application itself determines the required quality of filters in terms filter specs, and requireed processing throughput in terms sample rate $F_s$, samples per second.
High quality filters will tend to have tight specs, and that will require high orders for a given type of filter. Different types of filters, however, can meet similar specs at different orders.
The order of the filter, and the given data (sample) rate $F_s$, in return, determines the processing power requirements. For example, an FIR filter of order $N$ will perform $N+1$ MACs (multiply accumulate) per typical output sample for long input sequences; and if the data sample rate is $F_s$ samples per second, this means the filter will perform $(N+1) \times F_s$ MACs per second.
Given a CPU with a clock frequency of $M$ Hz, and MAC efficiency of $L$ MACs per hertz, then your processor should provide $M \times L > (N+1) \times F_s$, excluding any overheads due indexing, looping, or memory operations.
This is for a typical FIR implementation on single core, single thread system. There are a variety of architectures to implement FIR / IIR filters based on multirate techniques, or parallelizations based on SIMD techniques, which can affect the processing power.
Finally, as a figure of very rough merit, for audio applications at $44.1$ kHz, an FIR filter of order less than $20$ will have mostly weak properties. Orders between $20$ and $90$ will have (increasingly) mild qualities, and orders larger than 128 can have good to high quality characthersitics, where high quality refers to more control on the filter frequency response's spectral shape. Be warned that thousands of taps might be necessary to realize a sharp transition, narrow band, FIR filter.
In late response to comments below, it's possible to implement an FIR filter convolution drastically more efficient by using FFT based frequency-domain multiplication technique. My answer concerns time-domain implementation at this point.