Is it okay to run Apple's Thunderbolt Display in portrait mode? Will it overheat when used in portrait mode? (The air intake holes are in the bottom of the monitor when it is used in landscape mode, which provides good ventilation. When you put the monitor in portrait mode they're on the side. Will that be a problem?)
No, that shouldn't be a problem.  As long as the vents are not blocked, it should be fine.  I would only worry if you notice it getting hotter than usual, in which case I would put it back in landscape mode.