What is it called when someone is done with the task that he/she enlisted for?
- They enlist.
- They do what they signed up for.
- They __.
I would say they accomplished the task.
The etymology of the word enlist has to do with the making of a compact between one singular party and one collective party composed of members who contract en masse. Any word having to do with divesture, separation, or even nullification will suffice. The word 'fulfill' and its derivatives would be accurate in the context you describe.
The best word to use generally of oneself is resign, but there are a number of alternatives depending on context.
One antonym of enlist is delist, but that would normally have an object which might be reflexive ("I'd like to delist myself"), and as the ODO link shows, isn't normally used of people.
If you have enlisted in the armed forces, then you are demobilised (or "demobbed") at the end of your service; commissioned officers can resign their commission; members can be cashiered if they are discharged dishonourably.
Thesaurus.com has a number of additional possibilities based on resign, not all of which will be entirely suitable all the time: