I'm reposting my response in a thread over in the GM section (and I encourage any struggling with Obligation to review that thread):
1) My response is going to be pretty long.
2) This response will contain minor spoilers for the adventure, SHADOWS OF A BLACK SUN.
3) I have not GM'd EotE yet, so my response is entirely theoretical.
So...
Based on the description in the book, it seems clear that the game designers want Obligation to be a bit of a wild card, but not one that disrupts your game. It's meant to enhance, not disrupt, game play. In that sense, you should NEVER feel obliged to gut or even alter your intended tale because of the random results of a die roll.
Personally, I view Obligation as the potential for "story seasoning," or setting a secondary tone/"B-story." I do my campaign like a TV series (22 sessions per season, complete with a mid-season and season finale) and I see Obligation as a way to add another layer of things.
Consider a show like BUFFY THE VAMPIRE SLAYER. At its core, each episode is about pretty much the same thing: Buffy defeats the Monster of the Week. (Note: If you're a big Buffy fan, as I am, please don't call me to task on this oversimplification; I'm trying to make a point.) But depending on whose "obligation" has come up, how the episode plays out is going to differ: Are Xander and Anya arguing this week? Or is Buffy trying to figure out what she can do besides kill critters? Or maybe Willow is concerned that she might "fall to the dark side"? Or maybe Spike is particularly maudlin over Buffy... Regardless of the spin, the end result will be the same (Monster of the Week is defeated), but the path and tone might differ.
For GMs and players who relish the "role-playing" aspect of the game, I think this is a fantastic tool. Similarly, for a GM looking for some random "spice" session to session, there's just so much potential...
SHADOWS OF THE BLACK SUN gives some great minor manifestations of Obligation. They vary in detail, but all boil down to the same thing: If the PC's Obligation is activated, things get just a little more difficult for the PCs.
Character A's Obligation is Favor (basically, she works as a slicer for some criminals). If active, there's one scene where her attempts to get into a business's system are upgraded because high-end corporations know about her growing reputation are setting up counter-measures.
Character B has Family (both he and his brother work for the same crime syndicate). In the adventure, there's one particular social actions where the difficulty of skill checks get harder because the criminals the PCs are dealing with recognize the PC (and don't especially like/respect the brother).
Character C has Criminal (she's from a family of criminals). If her Obligation is active, security cameras pick her up at one point (assuming the PCs take certain actions) and move in to arrest her. Although this IS an additional encounter, it's relatively minor one.
Character D has Betrayal (a fellow bounty hunter as a minor nemesis). If this Obligation becomes active, the nemesis shows up -- but rather than a new encounter, he joins some criminals in an encounter that would happen either way. His appearance just makes the fight harder.
Note that none of these options are especially intrusive to the overall plot. Three of the four make a single encounter or challenge a bit harder. The fourth is a larger complication, but still more of a hassle than a true fight. I think that's an excellent way to approach things: don't disrupt your own plot, but look for some way to make things just a little more difficult for the characters, just once, during the session. (Note that in the SHADOWS OF THE BLACK SUN adventure, most of this complications impact the group as a whole, but the "offending" player in particular).
I think another good way to activate an Obligation without disrupting a story too much is incorporate it into something that was going to happen already. For example, in THE EMPIRE STRIKES BACK, Vader MUST get to Bespin eventually for the story to work, but perhaps Boba Fett wasn't originally involved -- until Han Solo's Obligation came up...