see dead on one's feet.
Examples from the Web for dead tired
He was dead-tired, had even hesitated over coming to the Institute of Insight tonight.Ham Sandwich|James H. Schmitz
Perhaps we were not aware that we should perspire profusely, and be dead-tired getting up and down the ladders?
Already they were dead-tired, for they had had little sleep since their long painful trek of the night before.A Company of Tanks|W. H. L. Watson
She did not understand, but, dead-tired, she hid her face against his arm again and fell asleep.The Law Inevitable|Louis Couperus
Urania was at the station and was startled to see Cornélie look grey and sallow, dead-tired, with hollow eyes.The Inevitable|Louis Couperus