animals deliberately hurt themselves when put in captivity, or pets who starve themselves and die after the loss of their master or other loved one.
I also know a story of a dolphin who refused water and committed suicide in a persons arms.
We can't know exactly the psychology of other animals and how they feel in relation to their own society, but suffice it to say, I have no question they feel a sense of no worth. Some may not know how to actually "commit suicide", but live their lives in a miserable depression.
The common theme is feeling alone. If you end up alone, you are going to have problems. Life is one organism. No one can do it alone..
Edit: found a great link
Do Animals Commit Suicide? - Softpedia