Can a 20 amp fuse be use on 14 gauge knob & tube wire for air conditioner? The 15 amp fuse breaker pops after use for an hour or two?

NO! 14ga wire requires 15 amp protection, not 20 amp. Over-fusing a circuit could very easily lead to wiring overheating, and that can cause an electrical fire. Try eliminating other loads from that circuit, or run a new circuit with the proper wire and ampacity.

Noting that it is knob and tube tells me you have a very old house. Insulation breaks down, wire connections get oxidized. How many watts does the AC draw, it should be on the device plate? 15 Amps may be satisfactory for the AC depending on the wattage but you may have other devices on the tlne too. Do not put a 20 Amp breaker in.

You should use 12 gauge wire.

All above is good advice. The least you shold do is a dedicated 20 amp circuit using #12's as mentioned.
It is cheaper than replacing everything(fire). The voltage was likely dropping as well, which is not good for the AC.

You should use 12 gauge wire.Was this that important that you needed to make this suggestion nearly three years after the OP posted?

Besides, it's old knob and tube. I have a feeling it is already in place.