I think this may be in the wrong section but no dogs usually don't kill themselves. I have seen some birds do it though, parrots get very attached and can live for 100 years once the parrots owner dies it can become depressed and won't eat. I don't know if its trying to die but they will starve sometimes. I have heard only rare stories like this about dogs

I don't know about consciously committing suicide but they probably could get so depressed that their actions could lead to their death.

For example, there was this documentary on chimps in the wild. This one particular "teenage" chimp was so attached to his mother that when she died he refused to leave her body for days, and even when the rest of the pack moved on he still stayed with the dead mother. They found him weeks later, dead next to his mother. He didn't hang himself or anything but he didn't eat, drink,...

I haven't heard of any real stories of dogs, cats,..but I'm sure it's possible. As far as them going through the actions of devising a plan to actually end their life, I don't know. I think humans are the only ones unintelligent enough to do that.