If my child felt excessively tired, and I didn't feel he could keep up with the class work, I definitely would not send him to school. I am in a special situation, though. I am a family physician, and my patients would not be appreciative if I would cancel office hours because my son has the sniffles. So, I sometimes will take him to the office with me, and let him work at my desk on a variety of projects I have there to amuse him in that situation. If he is really sick, I will bundle him up with some pillows and blankets I have there for that reason. I am fortunate that he has been blessed so far with good health. I think he has missed a total of 5 days of school since preschool because he was sick (compared to probable 15 or so for
Disney vacations and two more for oral surgery). On snow days, or the dreaded, two hour delays, I will also keep him with me at work. Usually, there is already office staff by the time I arrive for rounds, and somebody will watch him while I see patients at the hospital. (He is in second grade now).
On the other hand, if he has a high energy level, and doesn't act sick, I will let him go to school with a runny nose. I have been called by the school to pick him up for vomiting, when all he did for the rest of the day was play at my office. He was no sicker than I was--the school secretary has gotten smart, though. If my son runs around too much and gets hot after lunch, he will vomit. She lets him lie down in the office for a few minutes. If he gets up and is more interested in the fish tank, then he is sent back to class.
Incidentally, I only rarely take his temperature. I rely more on his clinical appearance, and only take his temperature if he is acting sick. Having a runny nose, but otherwise acting normal is not acting sick.