I'm currently running a Pathfinder game where one of my PCs is a Human, who in a relationship with an Elf. This Elf is currently about 170 years old, and the Human is 28 years old. The PC was wondering if there were any ways that her character could extend their life to match that of her partner.
This is a matter of story and should be treated like one.
The answer here is not to dig through books to find something like this. It’s to discuss your character’s goals withe the GM and state that your character if always on the lookout for things that might be able to grant this boon. Make the quest to get it part of the game. There could be long-lost relics, gods willing to offer it for completing some quests, or other options. Just like you wouldn’t look to find how to get revenge on the man who slaughtered your village in a sourcebook, you shouldn’t expect a sourcebook to answer this dilemma.
The problems with digging through books are twofold.
You probably won't find exactly what you’re looking for, you’ll probably find immortality, which means now the elf has the same problem you currently have. In 3.5 there actually was a prestige class that gave elf-like longevity to other races (ruathar, Races of the Wild), but that is not in Pathfinder. You could try to reincarnate into an elf, but that is really awkward—and not necessarily what your character really wants.
This is more systematic—the game’s rules only have one way to make something special: they have to make you pay for it. Immortality should be special (or else everyone would have it), so the game makes you pay quite a lot for it. But there is a mismatch between what you’re paying (metagame character-building resources, such as feats, levels, or whatever) and what you are getting (in-character narrative benefits, which in this case have zero metagame value).
The GM can handle this situation vastly better than a rulebook can, because the GM controls the entire world—and thus can require you to do things in-character, via the narrative, in order to get those narrative benefits. And he can and should do so without making you expend metagame character-building resources for it, because it has no metagame character-building value. It’s story, not character building.
This problem is actually remarkably similar to the problems with the Leadership feat (though that is definitely not at risk of being wasted resources; in that case, it is often overpowered). Leadership should be built up through roleplaying and narrative; you should have a cohort and followers because you are someone people want to follow and so on, not just because you took a feat. Likewise, you should have something rare and special like immortality because of some grand quest, not because you took a feat, mythic power, or class feature.
Because that kind of mechanistic approach just isn’t satisfying unless it gets worked into the story anyway. “I hit \$X\$th level, now I have immortality babe, we can spend eternity together now!” just kind of falls flat. Obviously you could (and should) do more to make it part of the story, but if you are doing that anyway, why bother digging through books?
Relocate to the Astral plane
Creatures physically on the Astral plane do not age, nor do they suffer of hunger, thirst, poisons or diseases. Natural healing doesn't occur either, so one should bring healing magic too. Permanently residing on the Astral plane would freeze aging completely and allow both the human and the elf to live together forever, but leaving the plane to one where time flows normally would cause the characters to resume aging.
If you'll be gaining in Mythic tier levels, you can get the first tier ability "Longevity"
Longevity (Su): Upon taking this ability, you can no longer die from old age. If you have penalties to your physical ability scores due to aging, you no longer take those penalties. You still continue to age, and you gain all the benefits to your mental ability scores. PRD
Perhaps something the PC can work towards in their adventuring career.
Take the capstone ability at level 20 for a Wizard or an Alchemist:
The wizard has the immortality arcane discovery:
Benefit: You discover a cure for aging, and from this point forward you take no penalty to your physical ability scores from advanced age. If you are already taking such penalties, they are removed at this time. This is an extraordinary ability.
The Alchemist has Eternal Youth grand discovery:
Benefit: The alchemist has discovered a cure for aging, and from this point forward he takes no penalty to his physical ability scores from advanced age. If the alchemist is already taking such penalties, they are removed at this time.
Ideally both characters should take this and then live together in harmony, awwww.
Alternatively, every time the human character dies just Reincarnate them. You can be reincarnated if you have died from old age and start all over again. This may put a strain on the relationship depending what they come back as, but then it might be an elf - so they could be deliberately and repeatedly reincarnated until they are an elf.
There are poisons out there that speed up the aging process. Therefore, the problem can be reversed: Shorten the Elf's life to match the Human's.