First thing to look for is a TR8 instead
I owned both. While the TR7 is a nice car, the TR8 is nicer. It rides smoother, has a more reliable engine and a little more power. Sorry if I offend....
Having said that. I would be looking at any body joint for rust. Check the strut towers and , as mentioned, the rear swing arms fro rust or any bending. If any steel is pulled or rusted away in the rear attach points of the shocks, or swing arms, run away.
Try and stay with a 1980 if you can. The tranny is a newer five speed design and we can pretend that BL actually incorporated some fixes in along the production path :wink:
If the tranny is making noise all the time be wary as they are kinda pricy to fix. If it is just hard to shift or grinds slightly into gear, it may just be fluid related. The GM ATF has fixed a lot of five speed ills.
Check the head gasket area very closely. The TR7 heads warped if not properly cared for. If it is overheating, check for head gasket issues first....most likely there is an issue there.
If you own an MG you already know about LUCAS so I'll save the detail on that except for one. There is a very good replacement plan for the alternators on these cars. So if the alternator is acting up on the car you are looking at....smile and work down the price. For $100 or so you can install a new Delco unit that works way better than stock and looks pretty close to original.
Good luck!