Do fighter jet pilots ever use autopilot?
I'm not in the military, so you could say I'm talking out my rear end here, but based on my experience as a pilot: yes, military pilots use the autopilot all the time.

Here's my reasoning, in order of most to least convincing evidence:


Punching the autopilot off takes about a fifth of a second.  The button is right there on the yoke, for crying out loud.  The time it takes to turn it off is not a good reason not to have it on.
The autopilot is a labor-saving device.  When you're in an aircraft (and particularly when you're solo in one) there are times when you'd rather be paying attention to your chart or setting up the FMS or stretching or eating a candy bar instead of actually flying the plane. Also, when Air Force pilots fly long repositoning flights, the autopilot reduces fatigue between in-flight refuelings (the Navy uses carriers).
Times when you might have to leave the designated path for a mission happen a lot less than you think.  Again, not in the military, I'm just basing this on the realities of fuel and the number of military aircraft out there.  Maybe if you're flying close air support for teams on the ground or something, but remember that the military is first and foremost a logistics organization - it's a machine for moving people and things from one place to another.  Most of the time, military aircraft aren't on a hair-trigger moment's notice ready-to-engage posture, they're going from one airfield to another for training or maintenance or something.
In most aircraft, you can override the autopilot if you need to anyway.  Again, never flown a military fighter aircraft, but based on what I know about flying in general, even if you're chilling in your F-22 (with the autopilot on) if you grab the stick and yank, the airplane will move.  (I would be interested to know whether this is true of Airbus aircraft. EDIT: Question answered after 4 years!)