Yes. You are morally obliged to weigh your desires against the anticipated effects of your actions. However, this isn't usually (for me) a matter of favoring other people over myself, it's more favoring long-term me over short-term me. I want to live in a better world, and that means taking actions that improve it.
Breaking up with someone gently and firmly now is far better than breaking up horribly after years of misery. For both parties, and for both of your future partners. Easy win.
Likewise terminating an employee who's not working out (when that's not likely to change) - it's a long-term win for you, their coworkers (even if they don't know it), and sometimes for the former employee.
Utilitarianism seems to indicate that the greatest good for the most people generally revolves around their feelings. A person feeling happy and confident is a desired state, a person in pain and misery is undesirable.
But what about taking selfish actions that hurt another person's feelings? If I'm in a relationship and breaking up with her would hurt her feelings, does that mean I have a moral obligation to stay with her? If I have an employee who is well-meaning but isn't working out, am I morally allowed to fire him? Or what about at a club? A guy is talking to a woman, and she's ready to go home with him. I could socially tool him and take her home myself, but doing so would cause him greater unhappiness than I would have felt if I'd left them alone.
In a nutshell, does utilitarianism state that I am morally obliged to curb my selfish desires so that other people can be happy?