The elves weren't here because of the Ring. They where there to prepare for the awakening of the humans and to teach and help them. Aragorn will still need strong allies. And there are still plenty of orcs and other enemies to fight. And some elves never left Middle Earth at all (except if they died of course).,
Anyway, after the destruction of the ring most elves left Middle-Earth. Celeborn and some other high elves still stayed a while in Middle-Earth (Celeborn even set-up a new woodland realm in Mirkwood) but after a while he was off to Galadriel as well. The only elves left behind in Middle-Earth in the 4th age, were some wood elves, probably lead by Thranduil. Some of these wood elves went on living in Ithilien, lead by Legolas. Legolas himself left Middle-Earth after Aragorn's passing. But anyway, most of the elves -safe Legolas- who stayed still never bothered with Men and Aragorn, as which was the case for centuries since the Last Alliance.
Remember what Galadriel said : "We must depart into the West, or dwindle to a rustic folk of dell and cave, slowly to forget and to be forgotten." The war of the ring is the last stand of the Free Peoples of Middle-Earth, after which the world will change and dominion will be granted to the race of Men. Elves and Dwarves will disappear (Elves leave, Dwarves probably get extinct due to their peculiar mating habits). The remainder of dwarves and elves who stayed in the 4th age, probably just clinged to their dwellings, although the power of the Three had been undone, trying to swim against the current of the flood of fate, but of course without success : they were forced to go, all of them or disappeared.
Indeed, there were a lot of battles to fight, and lots of Orcs to be genocided, but as in the 4th age dominion over Arda was given to Men, elves didn't bother (too busy with clinging). The wars were between Men anyway, Gondorans vs Easterlings, and for the remains of the Orcs, Faramir probably had a big part in hunting them down, as he lived closest to Mordor (the ones in the Misty Mountains would probably never leave their holes again).
(Now the Council has to find out how to add the 100 mithril prize to Virumor's account.)