Do companions drain fusion cores?

No, they do not.

I tried putting a fresh 100 charge core in a suit, and telling Preston Garvey to get in. We then cleared a raider compound, and traveled back to town. I told him to get out and inspected the core. It was still fully charged.

No, they don't use up fusion cores for Power Armor. I also experimented to see if this would work with the gatling laser, unfortunately it did not :(

Also something to be aware of, the armor does still take damage and can break.

you can even take the power core OUT and they will still use the power armor. Ive got piper running about in full armor with NO CORE IN IT

I’ve had Nick Valentine in a suit for quite a while, and he doesn’t seem to need fusion cores. Not sure if it’s because he’s a synth, but on advice I have avoided equipping him with a Gatling gun of any kind.