They definitely rust. Make sure you examine the underside carefully with the car on a lift. Good luck in your search.
H Gary, Please fill out your profile with general location as that will help answer your question a little more. It all boils down to location. Yes a TR can rust like any other car and the outer edges of the doors would be one spot. The biggest factor IMHO is if the car grew up in a humid region.
Bumps and scrapes on the underside of the chassis. The main rust areas IMO are the sill covers. Replaceable though. You will need to have the car lifted to inspect, they run from front to back of the main chassis area. I have a RHS original spare if anyone is interested. Phil
Rust on a testarossa .....would have had a hard life. The Testarossa unlike a 60s or 70s Ferrari would be undercoated. Albeit it, not like today cars, but had better protection. Plenty of testarossa's to choose from with 7,000 of them built. If you come across one with rust , it spent a lot of time in the rain, or outside, or both , pass and look for a good condition car. Plenty of them out there with loving owners.
Cars rust in the presence of salt. Rain or humidity alone doesn't do it. I have lived in very wet environments and you don't really see any rust. Go out to the Avenues in San Francisco. Rains the same in the whole place. At 10th avenue you don't see any rust, don't care how old the car is. Go out to the 40's near Ocean Beach and the fenders are falling off 8 year old cars. I used to work in the area. We used to joke about guessing the avenue by how rusty the car is. Salt air and the car rusts from top to bottom. Salt on the roads and it rusts from bottom to top. Point is as said before, lots of TR's out there. Nothing would get me to buy one with rust, any rust.
Make sure you put it on a lift and look at the underside. My old Testarossa had corrosion all the way along its exhaust system, springs (all 6), various mounts and associated parts underneath (sorry, I'm no technician). Nothing on the frame, though, and the paintwork looked pristine from above. My F512M doesn't have even a hint of rust or corrosion, visible or under any inspection, anywhere. The Testarossa was driven in all seasons in the UK (wet, salt), the M driven in the dry only in Spain (no rain, no salt). The Testa was a pain and expensive to get clean. I went into it with my eyes open but still...as Brian says, I wouldn't do that again.