All of the ones I sold requires the glass doors be open.
I just looked at the specifications for several remote control systems, and none of them warned against using them with glass doors closed. But perhaps these specifications weren't complete ----they had no warnings about exposure to high heat at all.
But consider: the electric gas valve is in there near the gas logs in any case. I never encountered a gas valve that was obviously hot or being overheated. If the gas valve isn't being overheated, I suggest that the remote control receiver is going to be OK as well.
And in particular, the gas valve and receiver are usually well off to the side of the gas logs and the gas flame. They don't get a "look" at the flames to be hit by radiant heat from the flames, but instead are in the shadow of the logs.
I never found that the temperature when glass doors were closed was high. And the practice was that the glass doors be left open to provide enough combustion air to burn the flames properly. If the doors were closed, the gas flame would often start coming out towards the glass doors themselves, to get a supply of oxygen to burn the gas. So I found that people habitually operated the gas logs with the glass doors open if they had them in order to get an attractive flame pattern.
In short, I never found remote control receivers or electric gas valves I thought had been damaged by heat from gas logs used when glass doors were closed.
Of course, your experience and practice might be different. If you have a make and model with that kind of warning I'd be glad to be able to read up on what it says, if you'd care to post that information.
Always glad to improve on my judgement.
I think the biggest and hottest set of gas logs I encountered was at the main REI (Recreational Equipment Inc) store in Seattle. The logs were about six feet long and that thing had a LOT of gas going in it, and BIG flames! It was right near the entrance to the store, and had seating for customers who often sat around and enjoyed the fire. It was operated by a millivolt gas valve and a wall switch.
This big fireplace had a big set of custom glass doors which were kept closed to keep people out of the fireplace. There was enough combustion air coming in so that the flames were fine.
I don't recall finding that the gas valve was ever even warm ----cooled by the combustion air coming into the fireplace I expect, plus the gas valve was off to the side and not exposed to radiant heat from the flames.
I'll have to see if the store has a picture of that fireplace....