An AC without a filter can work but is not efficient for even a few hours. If there is no air filter in your AC, it will harm all other air conditioning system parts.
Using an AC without an air filter or an AC with a clogged filter has almost the same consequences.
The after-effects of not replacing the AC filter are:
A filter that is covered with dirt will not be able to block the debris from entering your compressor and damaging it entirely.
Once your AC filter gets blocked, your electricity bill will never come down; instead, it will rise more.
A clogged filter leads to damage to the condenser and compressor. Both the damages will result in refrigerant leakage.
A dirty air filter will not be able to provide you with filtered air. The unfiltered air has fine toxic particles to ruin your health.