Wolverine's trusty healing factor is one of the coolest power sets in all of comics. With his healing abilities, Logan is the best at what he does - and often what he does is heal a slasher franchise's worth of grave injuries. The implication of these powers is clear - if Wolverine can heal from practically any injury, no matter how grave, is he effectively immortal?
Wolverine first appeared way back over the span of two issues - Incredible Hulk #180 and #181 - with Logan going toe-to-toe with the Hulk himself. In his first appearances, Wolverine's healing factor served a more practical end - since his claws sprout out of his hands, it figures that he'd have some sort of regenerative property to heal the ensuing wound. Over time, however, Wolverine's healing factor, combined with his adamantium skeleton, proved the popular X-Men character was more than just durable - and perhaps borderline immortal.
Logan's immortality is a well-established element of the comic books. Wolverine appears as a central figure in a number of stories, including Old Man Logan and Wolverine: The End and Here Comes Tomorrow. Most recently, Wolverine has been seen in House of X and Powers of X in the futuristic Year 100, where he and Xorn were among the last members of a mutant resistance. In most futures, Wolverine's aging process is superficial at best as, provided his healing factor is working properly, he still able to shrug off all forms of injury with little or no consequence.
This isn't to say Wolverine cannot die. In the grimdark future of Days of Future's Past, Logan died after being on the receiving end of a Sentinel blast, leaving only a smoldering adamantium skeleton behind. However, even this death was called into question when Wolverine was similarly charred to a skeleton while fighting Nitro in the aftermath of Civil War - and he still regenerated back to his original form. Even nuclear blasts do little to slow down Wolverine, as he has proved capable of surviving them. One odd stipulation, introduced in Wolverine #57, states that every time Wolverine dies, he must defeat Azrael, the Angel of Death in Purgatory in order to return to life. However, this has largely been ignored by most writers, presumably because it places unseen restrictions on Wolverine's healing factor. It seems the only way Wolverine can truly die is if his healing factor is compromised, as it was during The Death of Wolverine, which ended with, well, the death of Wolverine. However, as with most deaths nowadays, Wolverine's end was far from final.
As with most storytelling elements in a collaborative medium such as comics, Wolverine's longevity is often decided by whatever writer or editor handling the Canadian mutant at the time. However, given the severity of the injuries that Wolverine has recovered from in short order, it seems unlikely that Wolverine can be killed conventionally. Some have stipulated that drowning can potentially kill Wolverine, or at least, for all intents and purposes while he remains underwater. This was tested in the comics, but the virus that compromised his healing factor was introduced not long after, making its effectiveness somewhat murky. There's also the lingering question of what impact cloning Wolverine will have, since Logan is now, like most of the X-Men, a clone of the original with his personality and memories psychically downloaded. Since Wolverine appears to be alive far in the future, it seems this procedure was a success - but it does add a potentially thorny avenue for writers to explore in the future.
Ultimately, Wolverine's immortality extends only as far as his popularity in the comics. While that has never been in question in the past, times do change. Jonathan Hickman's revamp of the X-Men had drawn the Merry Band of Mutants into uncertain waters, and even the once-dependable film franchise is finding itself at a crossroads before their MCU debut. It's unlikely that Wolverine will ever die off completely, but given that he's apparently fought the Angel of Death for every new breath he's taken, well, stranger things have happened in the mutant's long history.