Agent 13

In the comics, everybody knows that Steve Rogers meets his (temporary) end at the conclusion of the Civil War. The responsible party is, apparently, Crossbones, who snipes him. But there is also one other shooter, the one who actually did it, a person Steve Rogers actually trusted "“ Agent 13, Sharon Carter, Captain America’s girlfriend.

Apparently, Dr. Faustus brainwashed her into killing the superhero.

Obviously, there is no Dr. Faustus in the MCU. However, the idea of brainwashing people isn't new at all "“ after all, Bucky Barnes became the Winter Soldier after HYDRA used this procedure on him. So what if Sharon Carter is the person inside that stasis tube? What if Baron Zemo has captured her and now is trying to get her on his side?

Bucky Barnes will most likely try to keep that from happening to another person, while Steve Rogers has all the more reasons for saving her (after all, even if they are not a pair in the MCU, she is the niece of his former love interest).

On the other hand, Tony Stark might see brainwashing the heroes as the only possibility of bringing them to his side. So he also has a reason for wanting to keep her in the tube. But this is a rather thin argument "“ so maybe he is doing it out of spite.