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.
First, check that you have the batteries installed in the correct direction in the remote. Some use both positive buttons up, some down, and some one of each.
Second, check that the remote and fan do not have a channel switch. These are usually mounted horizontally on the fan and are frequently labeled A/B or 1/2. If there is one, the fan and the remote must be set to the same value. It doesn't matter which one, use the one that works best in your home. Make sure the fan is turned off at the wall before changing this setting.
If that doesn't work, remove the lighting assembly and verify that the controls are connected in the space behind it. There are generally three screws around the edge of the plate where the lights attach to the fan. It may be helpful to have a second person safely assisting you. The colors there are generally white, black, and green that match the connections above. Red and/or blue may be used for the control wires in there as well.
Best Answer
So you have 2 switches one that control is the fan and one that control is the light on the same circuit. Just cutting the wire off is not safe on the unused hot at a minimum I would cover it with tape but would preferr a wire nut for better safety. Wiring the light if there is a rib or stripe on the wire with the writing it would be the neutral, since we can't see that examine which wire connects to the center of the screw in shell and the outside the neutral goes to the outside and the hot goes to the inside or center pin. With some fixtures I have found things were not to the standard so I will check to verify this. I have found some really nice older fixtures that the quality can't be matched today unless you win the lottery, make sure the power is off before connecting things up.