Skipper the dolphin, also known as Cathy, committed suicide in front of her trainer (who later talks about it in The Cove.) She was wild-caught and shipped to the trainer, and just gradually became more and more depressed. Finally she swam to the trainer, looked up at him, and opening her breathing hole underwater - thus drowning. There was nothing wrong with her, she just... Did it. http://www.huffingtonpost.com/vicky-collins/can-dolphins-commit-suici_b_592856.html
Supposedly dolphin suicide is extremely common in wild-caught theme-park dolphins (and related creatures). They just get to a point where they stop eating or just give up and drown.
I have also seen lizards starve to death (not my own, but other owners who don't know jack shit). Though that's not straight suicide. They aren't jumping onto sharp sticks or anything like that. But some species of pet gecko, and other types of reptile, will get stressed - usually through the owner giving them no places to hide or constantly messing with them - and then refuse to eat due to it. They then, should the stressing continue, just starve to death. And this isn't in a week, a gecko can starve itself for months before it finally dies. The golden gecko is a good example, as they are naturally stubborn. I adopted one wild-caught in 2005 and he refused to eat. I had to treat him for parasites and even after the treatment he refused to eat. I had to forcefeed him for 3 months before he finally ate on his own, and it took him about a year before he became comfortable enough to come out of hiding. That bastard was stubborn and just trying to die. Even now, as healthy and robust as he is, if something upsets him enough he'll go off food for a few days.
Some species of bird will commit suicide. It's usually related to depression if a mating pair should be separated, as some birds mate for life. One is killed and the other stays around the dead mate for a while, then either stops eating or just doesn't migrate and dies of the cold.