There's an earlier scene that explains what Teddy meant: the Ares program as a whole was at stake (emphasis mine)
TEDDY: Can the Hermes function for 533 days beyond the scheduled mission end?
VINCENT: It should. The Hermes was made to do all five Ares missions, so it's only halfway through lifespan.
ANNIE: But if something went wrong...
VINCENT: We would lose the crew. And the Ares Program with them.
This explains the actual lines between Teddy and Mitch
TEDDY: We're fighting the same war. Every time something goes wrong, the world forgets why we fly. I'm trying to keep us airborne. This is bigger than one person.
MITCH: No. It's not.
To put it a different way, the Hermes crew was clearly ready and willing to risk their lives for Watney. They left him there, and there was no shortage of guilt amongst the crew. That's part of why Lewis has to put things starkly before they vote to go back to Mars.
LEWIS: Not this time. This is something NASA expressly rejected. We're talking about mutiny. Which is not a word I use lightly. We do this together, or not at all. Before you answer, consider the consequences. If we mess up the supply rendezvous, we die. If we mess up the Earth gravity assist, we die. If we do everything perfectly, we add 533 days to our mission. 533 more days before we see our families again. 533 days of unplanned space travel where anything could go wrong. Something might break that we can't fix. If it's mission critical, we die.
Do you notice what she's not talking about? The Ares mission as a whole. Imagine the reactions of the rest of the Ares project (which was far larger than just 6 astronauts) if the Hermes had been lost. The flip side is that there's no doubt the Hermes crew would have been upset at not having been given the option at all (and Teddy had already kept back the knowledge of Watney being alive from them for a time). It would have, at least, damaged morale among the personnel to know that NASA was not being open with the people in space.
Teddy is trying to put the risk on Watney to save the Ares program. Mitch is trying to put the risk on Hermes to save Watney.