Yes, Mr. Bones commits suicide in Paul Auster's novel.
Although this is not directly stated in the novel, it is implied. When Mr. Bones' new owners, Dick and Polly, decide that they will go on vacation to Disneyworld, Mr. Bones is ecstatic until he realizes that they are not taking him with them. He laments the fact that he will be left behind; after all, his previous owner, Willy G. Christmas, had always taken Mr. Bones with him wherever he went.
It is not long after this disappointment that Mr. Bones experiences 'the first of several painful twinges in his abdomen.' Over the next two and a half weeks, the pain spreads to his haunches, his limbs, and to his throat. Although he does not exhibit any outward symptoms of any illness (like vomiting or diarrhea), Mr. Bones finds himself feeling less and less lively.
When he is finally dropped off at Dog Haven, he fails to recover any of his previous canine enthusiasm for new experiences, despite the fact that it is 'a four star rural retreat...designed to accommodate the needs and whims of the most indulged and demanding pets.' Mr. Bones finds himself oblivious to any of the pampering the staff subject him to; frighteningly, he eventually discovers that he can no longer eat or drink normally. In fact, his strength is weakening, and he finds himself either hallucinating or dreaming strange dreams about Willy.
Soon, Mr. Bones finds himself with a high fever; he vomits, and is distressed to find blood mixed in with half-digested meat fragments and yellowish matter. The owners of the dog retreat decide to take him to a vet, but Mr. Bones hatches a plan. When he is put into the car, he makes his way to the driver's seat; as soon as the door opens for the driver to get in, he jumps out and escapes into the woods. The goal is to get to back home; however, Mr. Bones is so weak that he only manages to walk two miles before he collapses. This time, he dreams that Willy has come to reassure him of his place in Timbuktu, that magical country where humans and dogs will be able to understand each other. In the dream, Willy tells Mr. Bones that he will have to 'kick the bucket' first before he can be admitted to Timbuktu.
When Mr. Bones finally wakes up, he finds that his fever is back with a vengeance; this time, he knows that he will never be strong enough to make the trip back to Dog Haven, even if Willy would have wanted him to. Upon hearing the sounds of the highway, he decides to make his way painfully to the road. After forty minutes of struggling through thorns, bushes, and uneven terrain, Mr. Bones finally sees the six lane superhighway. He decides that he will play 'dodge-the-car' to see whether he is still the dog he has always been.
'Dodge-the-car' then becomes a euphemistic phrase for Mr. Bones' suicidal mission. He knows that he is in no shape to avoid the fast cars and trucks. The last line of the novel tells us that Mr. Bones prefers to think of his mission as a journey rather than a 'vulgar suicide' mission.
With any luck, he would be with Willy before the day was out.