The relationship between co-founders is usually the single biggest risk to a startup in the earliest stages, it's certainly the most common reason for failure we see at YC. Having (unfortunately) seen a lot of co-founder breakups over the past couple of years, here is a list of some common causes I've seen that underly the final "we can't reach an agreement on anything, one of us has to leave" conversation.
Being the CEO
The most common reason for breakups is usually when more than one founder has aspirations of being CEO. It's true that titles are generally meaningless when a startup is just two or three people but I believe that being able to answer the question "Who is the CEO?" is important. In fact, it's probably a bad sign if it even needs to be asked. The counter example people often use here is Sergey and Larry but if you speak to people who knew the two of them from the early days, you'll hear that Larry was always the dominant founder. Over time, one founder usually emerges as the public face of the company e.g. Steve Jobs, Bill Gates, Mark Zuckerberg. Of course they're not the only ones doing the work but you should decide early on how important being this person is to you. If you find yourself having simmering resentment that you're not the CEO on day one, it's unlikely those feelings will dissipate over time.
Sometimes one founder will be more committed to the startup than the others, even if the equity split is equal. Perhaps one founder came up with the idea and recruited his friends, who wanted to try out a startup but aren't as passionate about the idea. This can turn out not to be a problem if the startup hits some reasonable growth (growth is usually a solution to such ails as long as it continues). If not, the "attached" co-founders might start wondering whether they really hated it as much at BigCo as they thought or start debating whether they should return to grad school. In particular if the "main" founder is seen as the public face of the company, the others may not feel as though their personal reputation/identity is intertwined with the startup and thus they have less to lose by leaving. An extreme example of this is when a non-technical founder convinces a technical founder to join the company and gives them 10% or less of the company. Such relationships rarely (if ever) work out.
Uneven equity splits
It's not necessary to have 50/50 equity splits all the time but there should be a good justification for moving away from it dramatically. If one founder came up with the idea, is investing money, has spent a decent period of time working on it, built the first version of the product and found an initial group of users then it can make sense. However if you're still at the idea stage and one founder has significantly less of the company, this can often cause simmering resentment. In particular this causes problems for companies that started outside of Silicon Valley and move here. The founder with less % will start realizing during conversations with other founders that 50/50 is the most common configuration and inevitably start questioning why that isn't the case with him/her. Remember that if the startup is successful, you'll be working on it for the next 5 - 7+ years of your lives. The majority of the work lies ahead so compensate accordingly.
People enjoy and value different things. To some, solving hard technical problems is more important than anything else. That's great if your company is e.g. Dropbox but not so great if you're starting Airbnb. If one founder places an emphasis on design above all else while the other treats it as an unnecessary luxury, there will always be constant tension between the two. You might argue that this works fine provided there are clear divisions of responsibility but such divisions are not so black and white at the early stages of a startup. Resources are finite and stretched, it's unlikely you'll be able to accomodate differing top-level priorities, which will seep into the prioritization of everyday tasks. There are broader versions of this e.g. different approaches to work/life balance, different personal weightings of goals such as money/power/status.
Lack of candour
This isn't as specific a cause as the others but I've noticed it a lot so it seems worth mentioning. Sometimes I'll see a team acrimoniously break-up one day over a seemingly trivial thing, despite having seemed calm and fine for months. It almost always turns out there was some simmering ongoing issue that neither founder brought up e.g. perhaps one feels like the other doesn't meet deadlines often enough. Not bringing up these issues immediately causes them to build up over time, where each incident by itself doesn't invoke full-out rage but one day something snaps . This often happens with shy hackers who don't enjoy dealing with confrontation. You're doing yourself a disservice if you don't bring up what's on your mind. If you feel like you can't, then you should probably question whether you have the right co-founder. I don't know if this is a recipe for fixing a relationship, it may be that this is just the sign of a broken one, but I do know that these things never just go away by themselves.
Have you encountered or experienced other causes?