"Time off" is... not really a thing that Felix thinks about. He is, in some ways, like a superhero - charged with protecting his world, always on duty, etc., etc. Luckily his world has rather fewer major threats than many comic-book worlds seem to, since he's one of about half a dozen people qualified (in his eyes, anyway) for the work, and two of them are raising a family instead of doing field work. So he's got a lot of time when he's not actively doing any protecting, but he's always alert for it.

Fluttershy... I'm less sure. She takes care of the Ponyville-area animals, which is basically a continuous thing for three seasons of the year, but she always seems to have time to participate in whatever her friends are up to. (Except when she's completely absent and unmentioned, but that's not especially common.) She certainly enjoys doing it, and has explicitly mentioned that despite the "awesome holidays", her favorite part of winter is the end, when she gets to see all the animals waking up from hibernation. All in all, I'd say she's closer to the second category than the first, but neither really fits her.