Putting on my contrarian hat, I have restored a number of 1940s-1960s TVs, and I haven't seen a lot of obviously heat-related problems in the older ones. Those TVs were designed to operate without extra ventilation and many of them played for hours every day, year in, year out.
If you visit the ETF museum in Ohio, you can see some of their sets playing for hours at a time. I haven't noticed extra fans or special cooling on any of them.
In some newer sets, late 1950s to early 1960s, I have seen issues like toasty PC boards around tube bases. A cramped set like a Philco Predicta might benefit from extra cooling, as would a color roundie whose flyback was questionable in the first place.
Old capacitors and resistors degraded over time, but this is a function of age (and invasion of water vapor, for caps) more than heat. NOS caps and resistors that were never installed have the same problems as those that saw use.
All in all, I think moisture is a bigger enemy than heat. Playing a TV a few hours every week, as you're doing, is better than letting it sit idle, since the heat of normal operation helps to keep things dried out and happy.
As Einar noted, an unfiltered fan may carry a lot of extra dust into the chassis. I have cleaned out old computers that were seriously clogged with dust. That stuff didn't grow legs and walk inside; the fan brought it in.
Just my $0.02.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html