One of the characters is trying to get information out of the other by threatening him with being put on the plane, which, it's implied, would take him to an extremely disagreeable destination.

(Unintentionally reduced spoilers due to having forgotten many details of the film.)

Yes. The staring man (Smiley, played by Gary Oldman) is looking for a spy in his organization. The man being stared at is worried that Smiley is going to kill him (by putting him on the plane). The audience feels the same anxiety by proxy, because it appears that the actors are about to be hit by the propellers.

Look again at the actors. Smiley never stops studying the others' face, while the other looks scared, nervous, shifty, and guilty the entire time. Even without context you should be able to pick up that Smiley is cool as ice, and that the other man has something to hide.