I've been intrigued by the idea of porting the Obligation mechanic from EotE into other games ever since our first campaign. Since 13th Age has been the other main game that our group plays it seemed like a good candidate, but I never really pursued the notion since OUT, Backgrounds, and Icon relationships are existing narrative mechanics that can serve the same purpose (though not as well), and I figured adding a fourth would just be too much.
Then I came across this adaptation of the Warlock for 13th Age, from the Mystic Theurge blog. The genius of the Warlock is that it's not a class per-se, but rather an add-on that really gets down to the core concept for what makes it unique - the fact that you've made a pact with a powerful being. The additional class feature very simply gives you a free 1 point Icon relationship (which you can invest more points into) with whatever being you've entered into a pact with. Most likely this will not be an established Icon. Any 5 or 6 is more or less resolved as normally, but with a supernatural flair reflecting the fact that it's a magical pact boon. When you roll a 1, though, your patron has an Agenda for you. As the blog entry states, this was based on Obligation from Star Wars, but obviously re-skinned to fit into the existing paradigm for Icon rolls.
I began to think more about this idea when I was creating my Druid (playtest version) for our current 13th Age game. His OUT is that he was the only survivor of a town swallowed up by a hellhole, but in the process he was infected with the blood of a demon (he's a Tiefling). He has a 2 point negative relationship with the Crusader because they want to hunt him down, as they consider him an abomination. He's always on the run, thus creating the perfect set-up for Obligation! Just interpret any 1's rolled as his Obligation being triggered.
There are a few things to keep in mind when using this mechanic, though. As-is, it's adding a strictly "negative" consequence for your PC with nothing positive to compensate for it, though a certain type of player (myself included) enjoys the narrative complications that crop up even if the character wouldn't. Some players might not necessarily view it as fair though, especially if others in the group don't have this extra burden. There are a few different ways of handling this that I can think of off the top of my head. Like with Obligation, the PC could gain access to additional resources. This is basically how the Warlock works, since you get an extra relationship die with which 5's and 6's can be rolled. This is a good, tangible benefit for a player with "Obligation." Alternatively, you might extend the benefit of the existing dice instead of granting more dice, so that a roll of 4 counts the same way as a 5. In this way the negative results of rolling 1 are countered by the fact that you expand your chances of rolling a benefit by 1/6 as well. In both of these scenarios there is an additional burden placed on the GM, however. For some GMs who have an easy time using Icon rolls already, this might not even be considered a burden. But in our current campaign we're rotating GMs, and it's becoming clear that everyone but myself is having quite a bit of trouble actually using Icon results. The GM tends to groan when a lot of results are rolled for a session, and adding even more results would only exacerbate the problem.
This can be ameliorated by trying to replicate the Obligation mechanic from EotE even more faithfully. First up, in Star Wars a PC can increase Obligation for the chance to get more starting funds or some extra starting XP. Give the 13th Age character with "Obligation" some starting resources, too! A true magic item seems like a good candidate, though it might be a bit more powerful than some groups are comfortable with. A handful of runes and/or potions might work nicely, though. Another hallmark of Obligation as it works in EotE is that you can voluntarily take more on during play in order to gain some immediate benefit. The classic examples are taking on debt to get some gear you can't afford, or agreeing to some future favor that can be called in (probably at an inconvenient time). The Icon mechanic can handle this, too, since one of their suggested uses is an event-based roll for dramatic events. Simply give players with "Obligation" ample opportunity to utilize event-based rolls. Along those same lines, for groups that want to handle it in as simple a way as possible you can simply balance out the "bad things happen on a 1" mechanic by saying those good things that happen on a 5 or 6, those tend to be better than they are for players who don't have "Obligation." Because of your special status as a thorn in that Icon's side, the enemy's of that Icon are apt to look more favorably on you, and are willing to go that extra mile to help you out.