Should a GM use a PC's Obligations if they haven't been rolled at the beginning of a session, or should Obligations only ever come up if they are rolled at the beginning?

I guess it would depend if it was part of the scenario's intrigue.

If the scenario is centered around it or strongly linked then, why the hell not?

In the other case i'd say yes as long it does not have any devious effect on the main plot (not counting the PC's actions of course).

Story over rules. ALWAYS story over rules. So if it makes sense/would be entertaining to take Obligation beyond the roll? Do it.

A character's source of obligation can come up whenever you want in a story. The roll represents, mostly, the STRESS of dealing with said obligation. So really the two are separate in the story (the character worrying about the obligation, and the obligation itself), but often it makes sense that the character's obligation would take front and center of the story for your session, either causing, or being caused by, your PC's stress over the situation.

This is where set pieces can come in handy. If you want, you can have a set piece put aside for each of your players' obligations. When the obligation comes up, the set piece also comes up (maybe it's an encounter, maybe it's a conversation, maybe it's just a cameo by a villain to whom the PC is indebted).

Thanks for the advice!

I figure that if a PC's Obligation is rolled it should be the one to come up in play. However, if the PC's are on a prison planet and someone has an oath to rescue someone from prison, that obligation should be able to appear regardless of rolls.

Yes and in fact the module beyond the rim bears this out as well as it has a number of side line encounters for certain obligations, that you can use, so you don't have to be stuck with just the roll. Some whole adventures are keyed to lowering your obligations. In fact Iuse obligation likw destiny points, as you get rid of some you might want to take on some more.

I usually roll Obligation for the NEXT adventure (so it is once behind). This way I can prep for an encounter.

I have several things happening in the background that the characters are unaware of and a group of encounters prepped. When an obligation is triggered, I then have until the next session to work in what is going to happen.

For the record, I have also delayed when this is trigger happens to because of one reason or another. For example, a finale or important main story thing is happening next session and it will not fit. I will make it happen next time.

I have not had this happen, but if obligations get "backed up" they will continue until used up.

Absolutely Obligation should be used beyond just the roll. In my opinion, Obligations should be used often -- and, whenever possible, in conflict with one another.

In my Saga game, the PCs essentially had Obligations, even though the idea didn't exist. For example, one of my PCs was a Jedi who became Sector Watchman. Another was a former slave pitfighter who created a guerilla organization to wipe out slavery. Often, their goals were compatible -- until Sakar (the pitfighter) wanted to cross a moral or legal line that Ni-Lah wasn't ready to cross. Hilarity ensues.

The roll, has been pointed out, is to inject a mechanical impact into the game. In the example above, if Sakar's number comes up, it may not impact the story at all: perhaps he's just stressed because Freedom's Hammer (his organization) is planning an operation that he can't supervise personally because of the main story. If doubles come up, maybe Sakar is SO distracted that it's impacting everyone -- or maybe he just picked everyone's brain in a late night planning session to help his people, so everyone is mentally drained.

Have fun with it.