Suffocation? His mutation pushes his body beyond its natural limits and he could probably hold his breath for as long as a dolphin I suppose. Blood loss? Perhaps his healing factor would constantly generate a copious amount of blood and there wouldn't be any problems of him having anemia. Starvation? Maybe he could shave off some flesh; maybe cut off a muscle, regenerate it, then cook it so he could eat it. Problem solved.
If you take a look at his page in the marvel database, you'll see it's not that simple. Even if you do find some way of putting the guy down (I like to think of him as a pissed off Energiser Bunny) Wolverine long ago earned the right to fight Death (or someone similar anyway) for his life whenever someone manages to inflict a killing blow/suffocates him/etc. Also, he once told Jean Grey that he'd survived beneath a glacier for six months by cutting strips off of his arm and having it grow back, so starvation seems unlikely.