The phrase "be in the running" typically means someone is being considered for something, such as a job or award. Antonyms for this phrase could include "eliminated," "excluded," "dismissed," or "written off." These words all suggest that someone is no longer being considered or has been removed from contention altogether. Other antonyms could include "out of the running," "out of contention," or "out of the picture." These phrases all indicate that someone is not being considered or is no longer a viable option for whatever they were initially in the running for.