I've run into this in a few garage door openers (and other fixtures as
well) - there's a small tab at the bottom of the bulb socket that makes
contact with the bulb. Over time that tab gets compressed and doesn't
spring up a make contact with the bottom of the bulb.

Make sure you unplug the garage door opener and then take a small screwdriver and bend that tab up slightly. If that doesn't fix the problem, there's probably something wrong internally with the opener. Could be a loose wire or a bad relay.