For me the weight would be an issue. However, you can use a standard TV wall mount to get around it. If you get rid of the exterior casing on the TV you can get a smaller “footprint” when paired with the mirror. Really though, like Sean said burn in will be your biggest enemy here. You could write something to have the modules swap places every now and then to try to prevent it, but given that there isn’t really much motion I think you’d end up regretting it. That said… it’s free. So, I figure you’ve got two options:

It’s a free TV the’s going to be a screen behind a mirror. How worried about it do you need to get… who cares right?

Sell it on Craigslist, Gumtree, eBay, etc and put the funds toward a cheap thinner LCD. That display is a 480 widescreen (from when people mistakenly thought DVD’s were HD and not SD like the VHS replacement they were… so 1995), so I don’t know how much you’d get for it these days. Regardless, money is money.