The brain, though a small organ, is responsible for a lot of things: It makes your body move, tells your heart to beat, and keeps you breathing without conscious effort. So it might make sense that using our brains for strenuous mental activity tires us out—when the brain is tired, so is the body.
However, the connection between thinking too hard and being worn out seems to be entirely in our heads. An experiment by Samuele Marcora of the University of Kent had some subjects play a mentally challenging computer game while others watched a documentary about trains or cars; both groups then took an endurance test on an exercise bike. Those who played the challenging game gave up pedaling sooner than the relaxed documentary-watchers, suggesting that the mental strain did, in fact, tire them out.
However, Marcora found no relation between the mental exertion and cardiovascular effect—the blood pressure, oxygen consumption, and cardiac output of those who had played the game all remained normal. The subjects who were assigned the mental task came into the endurance test perceiving it as more difficult than it was; their physical exhaustion from thinking too hard was a trick of the mind.