Yes, some do get themselves killed, however this is most probably not suicide.
In order for it to be suicide the animal would have to be aware that there is an alive and a dead and choose to be dead. Being as animals can't speak we know that some of the smarter animals can express grief due to the missing or inability to move (dead) cause them to starve themselves, but we don't know if they are choosing to permanently change themselves from alive to dead. All evidence shows that animals do not understand death as they don't have rituals or actions that are solely explainable by an understanding of life and death.
Elephants may disperse the bones of their brotheren, but this is likely instinctual to hide their migration patterns.
Dogs may jump off the roof of a house when their owner or lover die or go missing, but this may simply be a wish to leave the house or a cry for help.
In my opinion no animal other than humans have the ability to commit suicide because no other animal has the abstract intelligence to understand and choose death.