I have an old 1930's Shure 705A Crystal microphone that's intended to drive a really high impedance input.
Just to give some perspective on the impedance issue, this mic's datasheet gives response plots into 500K, 1M, 2M and 5M input impedances (nothing like a regular mic pre). The low end response is highly variant. From about 250 Hz to 7K this mic is nearly ruler flat (+/- 2dB or less), but below 250Hz things really start to happen. Into a 500K input impedance, it's about -5dB at 80 Hz and -10dB at 40 Hz (still pretty respectable). Into a 5M load it's only about -2dB at 40 Hz and maybe -3dB at 30 Hz !! This from a "harmonica" mic!
Of course nobody has anything with a 5M load, except maybe an acoustic guitar DI box.
Just illustrating that, depending on the mic in question, input impedance can have a pretty big effect.
FWIW the higher the input impedance, the more susceptible the cabling/etc. will be to picking up radiated interference. Properly balanced this shouldn't make a significant difference.