Electrical specification for minimum bias current available from 3.5mm cellphone microphone input?
(Not enough reputation to comment on @Kevin Reid's answer so I have to do so in another answer.)
The Android docs are a useful source of info regarding this question. I interpret them differently though: I think the 1.8 V minimum bias voltage in the spec is an open-circuit value. It's hard to be certain of course, because no output impedance for this voltage is stated.

However, the text also talks of a "2.2V mic bias applied through 2.2 kOhm resistor" which one might infer to be the reference method of providing the bias supply, and would allow for no more than 1 mA to flow (into a dead short). Given that mics which need bias tend to draw tiny currents, this would make sense.

Finally, the TI chip referenced in the question defaults to 2.0 V for the mic bias voltage and 2.2 kOhm for the mic bias resistance. This would give just over 0.9 mA into a dead short.