Ceiling fan light fixture not working
That's easy. The problem isn't in the fan at all, but rather in the dimmer switch (on the wall, probably).

Most modern dimmers use a kind of transistor (a triac) to regulate the amount of power (and therefore brightness) flowing to the lamp. Those triacs do fail eventually, and when this happens they normally fail "open" (off).

These dimmers will still allow a tiny bit of current through. Not enough to light a single light bulb, but enough for a sensitive voltmeter to detect. If you disconnect the light bulbs and measure the voltage, you'll get close to the normal 120 VAC; but even a single light bulb will "sink" the tiny bit of current leaking through the dimmer, and the voltage drops to near zero. (You could have done the same test by just unscrewing the light bulbs and testing the contacts in the bulb sockets).

Note that for some electronically controlled fans, the dimmer electronics can be in the fan housing and not in the wall switch. This would be particularly true for fans with a wireless remote.

Replacing the dimmer will almost certainly fix your problem. (If you want to test the hypothesis before buying a new dimmer, you could bypass the dimmer, which will, if everything else is good, result in the lights being "on" all the time).

Obviously, make sure the breaker is off before doing any of this.