1. Near to the point of collapse or losing consciousness (as due to exhaustion or injury) while still remaining on one's feet; just short of being asleep or unconscious. Doctors in training are expected to endure an unbelievable amount of stress and exhaustion. Surely it is counterproductive forcing them to attend to each of their patients while they're dead on their feet.

2. Still functioning, but past the point of usefulness or productivity; All but or as good as defeated. The company has managed to remain open, but, truth be told, it's really been dead on its feet for the last year.