They probably would, to a degree. I think UV rays are a large contributor to dryrot in wood; not so much the heat. I'm reasonably comfortable assuming that heat under 110 or so shouldn't be a problem, judging from the fact that guns have operated in non-air-conditioned environments for a very long time without any mishap. It'd have to get much hotter than 110 to cause any real damage, I think.
Also, being as a gun save is big and metal, and garages are often made out of cement, you might get a couple degrees of passive conductive cooling from the ground.
But seriously: why not put the safe in the garage or in a closet? It's your house too (and, I'd suspect, you're paying for it moreso than she). Don't you have a den, or at least a personal study where you can do what you wish? It's only reasonable if she's got the rest of the house to 'decorate' as she sees fit. Or does she expect you to be out working during all your waking hours? Put your foot down! If you don't respect yourself, how can you expect her to?