What would a voltage between 0 & 1 do to a AND-gate IC?
The datasheet for the IC will define "threshold" voltages. Below the lower threshold (e.g. 1V) will be treated as zero; above the higher threshold (e.g. 2V) will be treated as 1. Normally there will be a small gray area in the middle between the thresholds. Depending on the IC's construction (TTL or CMOS), it may turn on both the "0" and "1" output transistors, resulting in high power dissipation and possible damage.

"Schmitt trigger" inputs can be used if you want to quantify a signal that is wandering about; these will stay at 0 or 1 until the input is almost at the other end of the voltage scale, then flip over and stick there.