Garage Door Opener Doesn't Work with Remote When Light Is On
The RF communication is degraded.  Energizing the wire to the light merely increases the noise, masking a weak signal.

The problem could be a defect in the remote(s) or a degradation of the receiver circuit or both.

Here's what to do:


First make sure that the remote batteries are fresh.  Use a battery tester (not just a voltmeter), or just replace the battery.
Double check that the receiver antenna wire (if any) is not mangled and has a good connection internally.
If the problem persists, it is most likely the receiver/control module.  On a unit that old, a variety of failures are probable, especially electrolytic capacitors.  Dirt or smoke residue can accelerate failure of IC's and transistors as well.
Given that the unit is 20 years old, and uses belts and plastic moving parts, it's about due for mechanical failures anyway.  Replace the garage door opener.
Make sure that the wiring did not contribute to premature failure.  Check all AC power to the opener for proper hot/neutral/ground configuration.  If you suspect spikes or brownouts, have that checked out too.