In The Adventures of Dr. McNinja, Dracula thinks he's Seen It All and is interested in dying. However, he's not stupid enough to die without knowing what comes next, so he comes up with an elaborate scheme to perform reconnaissance on Purgatory.
Atomic Robo: Robo has spent over a century fighting mad scientists, natural disasters, entities from beyond time and space, and secret government organizations that hate him. When a newscaster asks him what has been hardest, he sadly says that he's just old. He's lost a lot of friends, time has moved on, but he's still here.
Robo: I do a great Jack Benny, but no one really gets it any more.
CharCole: According to legend, Maoh the Mew was cursed with immortality for trying to rob a shrine. And according to Crop Town's gym leader, living forever is sure to bore one out of their mind.
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In Deathless, a rich man discovers the secret to immortality, but decides not to share it with the world until he feels the time is right. He ends up regretting it when the world falls apart and humanity goes extinct, leaving him the sole survivor.
Quain'tana of Drowtales can feel the centuries creeping into her body and knows she will soon never be able to venture the world again because the lack of mana would quickly cause her natural immortality to cease and her to age. She hopes she will be able to venture the world again before this happens, but fully plans to go through with it even if it does kill her. She knows this all too well because the Empress Diva'ratrika was so old that she was unable to leave her tower, and most elves old enough to experience this wind up committing suicide out of despair.
In 8-Bit Theater, Sarda avoids this for the first two weeks of the universe by growing a mustache, but quickly reverts into "Blind, seething rage."
As Immortals in El Goonish Shive age, they slowly get more intelligent and powerful. Eventually, they get to the point where they're driven insane from boredom because they're practically omniscient, and eventually commit suicide and take a ton of innocents with them. To avoid such a faux pas, most Immortals "reset" every 200 years, wiping their own memories and starting anew.
Here it's discussed by an elf with a mayfly who laments having a very short life, while elves live forever. The elf replies that the mayfly shouldn't be sad in the end, because immortality runs thin. Over time, you do everything and see no point in doing things. His people are burdened with massive boredom, always fearing they will run out of something to do. The mayfly therefore dies comforted by the idea that a short life is better. However, then it's subverted as we learn this was just a lie—in fact, the elf is still working on new projects, with no sign he actually ran out of things to do with his immortal life (although you might interpret this as the Elflord going back to work after the Mayfly shows him not to take life for granted).
Another comic, deriving its plot and title from a novel by Simone Beauvoir, explores the same idea with an immortal king trying to change humanity for the better but failing miserably, which causes him severe depression. It's stated he may someday be left alone on Earth, with everyone else dead.
Exvulnerum boasts The titular curse, which not only gives the victim immortality, but causes them to feel the pain of every living being during the night. sometimes even worse than the living being itself. The only way to get rid of it is to transfer it to the person they care about the most.
This is a major plot point in Ginpu, where demigods are doomed to spend their lives losing those they love. The BBEG's plan revolves around making a child who is immortal, so he'll have at least one family member that doesn't die.
Guess what Morgan's biggest desire is of God? Live, grow old with family, die like a human.
In the Goblins spin-off Tempts Fate, a demon Tempts has destroyed curses him with immortality so that when the demon's body regenerates 10000 years later, Tempts will still be around and the demon can take revenge. Tempts, being a thrill-seeker, is initially horrified by the idea of being unable to die, although when he learns he can still die from serious injury, his attitude shifts to Cursed with Awesome.
Homestuck: The Handmaid, due to being at the bottom of Alternia's caste system should have died naturally at about the age of twenty-six, but is cursed with immortality by Lord English. The only reason she follows his plans is because at the end of her service, she'll finally get to die.
In Sburb, an overwhelming majority of all sessions fail in every timeline. This does not mean doomed timelines, which will kill everyone inside of them eventually, but their Alpha timelines. We have no reason to assume that everybody who played Sburb on Earth failed to enter the game, meaning that there are some sessions of Sburb out there that have trapped children inside of them. Now, the Fridge Horror in this is bad enough, but for the people that achieve God Tier, it is the curse of immortality. There is no way for everybody to get Mercy Killed, as at least one person is going to be left, and suicide is neither a Heroic or Just death. They will spend eternity in a game that will kill them over and over and over again, with revival certain every time.
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Martin from Ignition Zero is a Half-Human Hybrid. He is immortal while his friends are not. He notes that being aromantic he won't feel the loss of a lover, but seeing all his family and friends die while he stays forever young isn't any better.
In The Inexplicable Adventures of Bob!, the Nemesites are not immortal, but do have lifespans upwards of a thousand years. That's not a problem for them so far as it goes (indeed, it's what makes their interstellar empire possible in a universe without faster-than-light-travel), but it does mean they outlive any friends they make from other planets. Voluptua is very unhappy to realize how brief Bob's life will be compared to hers.
In Jack, it's been revealed that it's possible by a sort of cosmic accident for people to "miss their deaths", and thus achieve an immortality of sorts. The drawback is that it's a stasis in which they don't need to eat, sleep or even breath except out of habit, they can't grow or procreate, and they can't progress or improve.
In Jix, Kelelder the Planet Thief got bored after immortality was thrust upon him and started killing his own kind and claiming their colony planets as his own.
Edermask from Magician hardly angsts about his immortality but he is looking for a way to grow old and die naturally.
Discussed in this strip of minus.. The three kids are talking about how being human is the way to go- magic would make things too easy and living forever would take the satisfaction out of life... then they see Minus floating by on a cloud, and silently watch her from the ground. It implies this trope is just sour grapes about our own mortality.
In Raven Wolf the titular tribe was cursed by their totem spirits with "removal from the cycle of life" until the domestics (a faction of "civilized" furries) are no more. The usual angst about outliving one's loved ones is averted because anyone who marries into the tribe is cursed as well and their children inherit it, but if they fall in battle their souls are devoured by the wolf spirit. Also they can't hunt, gather, or cultivate, they depend on the charity of others for food.
In the Back Story of Rice Boy, a being who identified itself as God gave a mission to three people to find and nominate a Fulfiller, with the promise that as long as they continued searching they would not die. Many centuries and many false Fulfillers later, one of them has abandoned the mission and resorted to prolonging his life unnaturally with the Black Spirit. The second commits suicide by abandoning the mission, knowing that it will kill him. The third, after seeing his friend killed, finally despairs and asks God to kill him as well; but God has something else planned for him.
This strip points out the one consequence of immortality that simply can't be spun in a positive light.
This comic shows another little problem with immortality, educational requirement inflation.
Here, everyone getting immortality led to weird societal change where everyone was being sued for infinity money... Well, it's kind of hard to summarise.
Also: It would suck because you'd have to keep having sex with more and more people at once until you reached the largest possible number and then there would be no point.
Averted in this comic, which mocks the idea of this trope by picturing how people who actually obtain immortality will look back at it.
One race attempted immortality through technology, only to have it backfire on them rather badly - as their people invariably went insane after a few normal lifetimes as their mental health didn't regenerate like their bodies did - almost destroying their civilization. The few survivors altered themselves so as to live in a permanent state of senility to prevent something like this ever happening again. The whole ordeal is described in more detail here.
Interestingly, they're not the only ones. Humans and other races have a very top-secret longetivity project going on, and a member of the above-mentioned race is actually helping them- he figures they might as well try to get it right this time.
Deconstructed in a different direction later: After regaining their full intelligence, memories, and lifespans, the practically-immortal oafa work to perfect and distribute the immortality project Terrans had been struggling with for generations, in order to have actual peers.
Squid-Sophont: The Plutorialment will question your motives. Oafan Ambassador: They are a deep breath too young to understand our motives. Squid-Sophont: Old age makes you generous? Oafan Ambassador:Longevity is a curse if one has no friends with which to share it.
Zombies seem to have this deal. For a while they seem content enough (though having to eat human flesh to keep from decaying away to nothing must've taken some adjusting), but once you're reduced to being a Zombie-Head-on-a-Stick...
Done more seriously with Oasis.
Oasis: I love Torg so badly, the thought he won't return that love makes me want to die. And I can't die!
ThisSomething Happens strip features a commercial for an immortality drug, but the Side Effects Include... disclaimer mentions the unpleasatness of still being around as the world dies.
Unsounded: Prakhuta was experimented on and ended up fixed with an artificial equivalent of a Kassylinain soul. He wants nothing more than its destruction and finds the idea of it surviving past his death in the khert and possibly allowing him immortality through reincarnation unbearable, and is more than happy to commit genocide and collude with a god of evil to attain oblivion.
In Weregeek, Mark's character in their Vampire LARP thinks that the immortality is overrated, as it means dealing with vampirepolitics for all of that time.
Mark/"Prince Allen": Holy crap! Is everyone out to kill me?? This whole undead immortality thing is totally overrated!
, a rich man discovers the secret to immortality, but decides not to share it with the world until he feels the time is right. He ends up regretting it when the world falls apart and humanity goes extinct, leaving him the sole survivor.