The microphones of most smart phones or portable PCs are able to reliably record frequencies between 100Hz and 10kHz. Although a piano can produce bass frequencies well below 100Hz, a useful tuning software does not need to consider them as it can derive enough information from the overtones with higher frequencies. So it depends on the piano tuning app you are using whether the quality of the microphone plays a role.
There are several programs available which use different approaches for measuring and calculating the actual tuning curves:
Entropy Piano Tuner and Dirk's Piano Tuner try to minimize the total entropy of all overtones sounding together. Their developers state that an external high-quality microphone is essential to capture all overtones.
Pianoscope, PianoMeter, Verituner and TuneLab are optimized for smartphones and tablets and do not require the microphones to pick up low frequencies. They achieve a high precision by focusing on higher pitched overtones. TuneLab respects only a single partial per note whereas Pianoscope, PianoMeter and Verituner use a combination of multiple partials.
The actual linearity of the microphone's frequency response does not play a role for the actual tuning process. But Pianoscope and Pianometer additionally measure the intensities of the first ten overtones. From them they derive which tuning intervals to prefer in different parts of the piano range when they calculate their tuning curves. These calculations can be affected by a microphone with a non-linear frequency response. But in practice the variations in the resulting tuning curves are small.
Disclaimer: I am the developer of one of the listed apps.