Yes, Wolverine can die. His healing factor has grown over the years to the point where he was able to survive being caught in an atomic blast and having all the soft tissue incinerated from his body, yet within a short amount of time he was back to normal. His healing factor also prevents his growing old, since his cells don't age, therefore don't die. The comics also have him as being the victim of a gang of black market organ harvests, who removed all his internal organs, by reaching from under his ribcage rather than through it, and he was able to regrown his heart, lungs, liver, ect, before he was able to die. It is unknown if he can drown. While it is believed that decapitation can kill him, this is nearly impossible due to his adamantium skeleton. Also if he was decapitated his head would have to be removed from the vacinity of his body. Deadpool, a character who was surgically given wolverine's healing power, can and has survived decapitation, but needs to have his original head put back on his body for the neck to heal to two sections of him back together. Since his healing ability is based off of Wolverine's, one can assume that Wolverine might have this ability as well. His healing ability also makes Wolverine immune to death from any type of disease, poison, or toxin, though a srong enough dose can make him sick until his healing kicks in.
Since Wolverines healing factor prevents the adamantium in his body from poisoning him, one way that could kill him is if he came into contact with someone like the character Leech. Leech's mutant power is that he can make other mutant's powers not work. It is reasonable to believe that if you used Leech to take away Wolverine's healing power, then he would be prone to death from anything that doesn't involve getting through his skeleton. Other factors that have occurred throughout the comics can also turn off Wolverine's healing factor. According to "X-Men", Rogue's powers could also kill Wolverine, by extracting life energy and his powers out of his body. Magneto could probably kill him by meddling with the adamantium in his body.
In the comic series called Gaurdians of the Galaxy, which is set about 1000 years into a possible future, the skeleton of Wolverine appears, but Wolverine himself is dead. In this future, Dr. Doom turned Wolverine's skeleton into a robot into which he transferred his own brain. There was a continuation of the Origins story called Wolverine: The End which take place approx. 200 years in the future. Wolverine has aged to the point of being considered an old man (looking about 65-70) His healing factor no longer works as good as it used to, and the adamantium has even begun to break down, most notable the center claw on his left hand is actually broken.