After seeing someone complain about people telling the they look tired on Twitter, I had a little epiphany, saying, "You look tired," is like saying, "You look pregnant."
It is generally established that telling someone they look pregnant is not a great idea unless you actually know they are pregnant* because either: they are but you have now stepped on their announcement, they are but they didn't want to tell you yet, or they aren't and now you have made them terribly self conscious about their waistline.
You look tired (or it's friend - you look sick) is really the same thing. Now, possibly you happen to know that this person was up all night with a teething or ill child, went to a midnight showing of a movie, and/or was partying til the break of dawn. Or maybe you don't and now you have made that person terribly self conscious that they look old, or tired, when up until then, they thought they were having a pretty good day. And if they are in fact tired, you have burst their bubble about how well they were faking it.
I recognize that generally, you look tired is meant in a sympathetic manner. And I'm not trying to tell people to stop being nice. I am suggesting that this is not the way to go.
*Except, of course, for that one woman from "I Didn't Know I was Pregnant" who says she wishes her co-workers had told her she was looking bulky about the middle.