If you are logical and people know it, they can exploit you.
It's a cliche in movies that the bad guy will say something like, "Now, don't do anything stupid, and we can all walk away from this." That is, the person taking advantage of you is counting on you being logical and doing what's best for you.
Suppose you plan to rob someone at gunpoint. Would you rather try a logical person or a person prone to bursts of wild anger?
The logical person is the better target, since they will simply give up their wallet, then later call the police. The angry man could go off and attack you. That's bad for him because he might get shot, but it's bad for you because you might get hurt, or you might run away, or you might shoot him and the police will put some actual effort into tracking you down. If you're an illogical, angry person and you can signal it, you're less likely to get robbed.
Maybe this feels artificial, but I think it is not so uncommon that being perfectly rational could be to your disadvantage. You'd be more susceptible to blackmail and more likely to back down from a standoff. Ever watch a logical person play chicken with someone who just shot up on cocaine? Logic loses every time.
Logical people have eaten the forbidden fruit.
Sometimes your payoff may actually depend on your knowledge. Maybe believing in Santa Claus is valuable to you because you love the feeling of magic. Your goal isn't to get presents or to see Santa at the mall or to read Polar Express, it is actually to believe in Santa Claus your entire life.
You may well find that the act of logically considering how to achieve this makes the thing impossible!
Prisoner's Dilemma, etc.
The prisoner's dilemma is famous game in which logical players come out badly - they must betray each other to do what is best for themselves. The solution? Don't be logical. This, of course, hurts you when playing the prisoner's dilemma. Making the illogical choice is bad for you by definition.
The real solution to the prisoner's dilemma is not to be illogical, but to play with an illogical person. They can sacrifice themselves, helping you. If people know you're logical, they won't want to play a prisoner's dilemma with you. They won't even want to put themselves in a position where they might have to, so illogical people may have more opportunities.
The exact prisoner's dilemma structure is not essential. For example, suppose you have to choose a mate. Do you prefer someone who will sacrifice their career, their time, their money, etc. in order to raise the children as best they possibly can, even if doing so is draining and unrewarding? Or else do you want a heartless logical bitch who will just tell you to hire a nanny, then cheat on you with the pool boy?
Emotions are a signal for strategic irrationality.
What is anger anyway? It is a signal to the other person, "I am going to start acting illogically now. You had better rethink your strategy because I'm about to disregard my own interest and thereby become really dangerous."
Something similar applies to other emotions. Depression. Say you don't want to spend time with me because you're busy. If I'm rational, I try to induce you into spending time with me by creating an incentive for you. But it's much easier just to start showing depression, acting as if I might kill myself soon if someone doesn't come cheer me up. Of course suicide would be completely irrational, but I'm not acting rationally; I'm depressed. Recognizing this, you come give me attention, trying to cheer me up.
How do you really win the prisoner's dilemma? Play with someone for whom you feel mutual, wild, illogical love. How do you keep people from challenging your authority or trying to steal your stuff? Go into flights of periodic, self-detrimental rage to let them know you won't take their shit. How do you believe in Santa forever? Develop an overpowering illogical emotion called "faith".
The emotions arms race
All of the above benefits of being illogical are based on the knowledge of someone you're interacting with - they need to think that you're illogical for you to get a benefit. The best course of action then, is to be logical, but to fake being illogical. Pretend to be in love with the 70-year old tycoon, marry him, then stab him in the back and run off with the sexy young stud.
To prevent this, people using emotion to signal their state of illogicality should make the signs progressively more subtle and harder to fake. As generations go by, there is an emotionality arms race. On one side, there are fakers getting better and better at signalling false emotional states. On the other, there are emotional people getting better and better at making their emotional tells into subtle, subconscious cues, and also getting better at picking up on them. People who fall behind in this "emotional intelligence" battle wind up penalized in all their interactions throughout life.
We can imagine the humans resulting from such an arms race. Knowing who you could trust would be a key part of their lives, and they would have circles of companions, going from closest friends out to mere acquaintances. Managing their own emotional lives and their display of emotions would become their greatest challenge. They would constantly keep track of the various relationships between people, feeling almost compelled to gossip. They would also track other people's personalities and goals. They would use that information to judge the people they knew, forming images in their minds to emulate or predict others' actions. They might even come to enjoy this sort of thing so much they would make up imaginary people in imaginary scenarios and tell each other stories about what those people did. In this world, the experience, expression, and communication of emotion would be the pinnacles of people's lives - the key component of their society and art.
They would be illogical, and better for it.
I borrowed heavily from Someone anonymous's answer to In Star Trek, the Vulcans believe that emotions lead to irrationality and should therefore be totally suppressed. Are they right or wrong?, which in turn seems to have borrowed from here: http://www.scottaaronson.com/writings/selfdelusion.html