My piano tuner has advised me to not use my store bought wool string cover. He says that it is overheating the plate and causing an unstable tuning. Does this make sense to you?

Sorry--I don't know the answer to your question, but I have my own if you'd be so kind. What are the advantages in having a string cover (i.e., why do people buy them)? (I'm about to get a grand for the first time.)

People use string covers to keep the strings, soundboard, plate, and all of the nooks and crannies inside a grand piano from collecting dust. String covers have pros and cons, they can keep the inside of a piano looking new for years, but so can closing the lid when you're not playing. They also mute the sound somewhat - which could be a good thing but it depends upon the circumstances. I'm not sure about the "heating up the plate" claim. That sounds doubtful. You can google "Edwards String Cover" and find information, but keep in mind that what you read at that sight is a sales page, meant to generate sales. Personally, I didn't like the sound of my piano with a string cover so I took it off. Your milage may vary.

unless your string cover is an electric blanket or your piano is sitting next to a heat source (sunny window, heat vent) it would have no effect on the temperature of your piano. if it's the latter, then the location of the piano is the problem, not the string cover.

I have two grands, both with string covers. The only reason a plate may heat up would be from under the piano where an older model Damp Chaser system that did not have a functioning thermostat control to turn it off would keep heating-then I suppose a string cover would keep some of that heat trapped and that could interfere with stable tuning.

Yes, the string cover keeps things clean and tidy. It does somewhat modify the sound you hear. In my situation I have large grands and a small house - string covers are a good thing. From time to time I take them off, shake them out and then sit down and play enjoying the full tone of these great instruments with their strings naked.