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.