are you trying to get power for the lights from the hot wire on the switch loop? thats not ok nor is stealing a random neutral. what you are proposing is about 10 different kinds of wrong. if the fan circuit runs to the fan first, which it sounds like when you say "There is a dedicated 20 amp
circuit that runs straight to the fan.", you can run a 3 wire from the fan to the rocker carry the hot down and 2 switch legs back up, and then it would be OK to use that neutral. I'm assuming of course that your rocker switch is capable of multiple feeds via break off tabs which most are. that could also explain why your current rigging doesn't work.

Right - I'm trying to power the bathroom mirror light, the fan light and the fan night light with a hot wire from the power that comes into the bathroom mirror light box. The 3 rocker switch only has one common on one side, and then 3 separate terminals on the other side. So it cannot take multiple power feeds. If I run a neutral wire from the bathroom mirror over to the fan and hook the neutrals from the fan light and fan night light to this, I think it will work. But I'd like to know why the lights won't even light when they are hooked to the neutral on a different circuit.

"But I'd like to know why the lights won't even light when they are hooked to the neutral on a different circuit."
I will not even try to explain. This is a hazard to anyone working on this system in the future. A code violation. I have been severely shocked a couple of times by such hookups. I will kick the a-double-rear of anyone I catch doing such a foolish and dangerous hookup.

Bus Driver - Ok, I get that it is dangerous and a code violation and I need to run a neutral back to the other circuit. But isn't it still technically a complete circuit and shouldn't the light still light? Could you please explain why the light won't light? I'd really like to understand why.

How did you determine the wire is neutral?

You are talking about the one I first hooked the lights to, the one in the fan housing? I know there is a cable that runs from a 20 amp breaker right to the fan housing. I first hooked the white wire from the fan light and the fan night light to the white wire of that cable, and it didn't work. I've since run a neutral from the fan housing back to the light above the mirror, and I have the fan light and fan night light white wires hooked to it. Everything now works. Why didn't it work the first time?

You can't just "run a neutral" The neutral has to be in the same cable with the hot conductors it corresponds with. You would have to run power to the switch first and then run 14/3 or 12/3 to the fan connecting to the neutral and hots for the nite light and regular light. Since the switch can not be divided, you can not run a separate loop switch to the switch. Why it didn't work with your other botched together scenario, I have no idea. In theory it should still light up, perhaps some other variable was in the mix like an unconnected hot. I hate to think of what the installation looks like with a home run entering the fan, a loop switch for the fan, random hots coming from a stacker switch to the fan, a power out feed to the gfci, and a random neutral coming from the vanity light. Forgot about box fill rules and the code on requiring all conductors to be in the same cable or conduit. Not how things are done, you don't just run single conductors around connecting to things until the lights light up.

Once again, this proves that just because you touch two wires together and the lights come on, does NOT mean it was done right.