Thor is supposed to be a god, so he should be immortal. Does this mean that there will come a time when he is the last remaining Avenger?
No, There were plenty of immortal Avengers over the years. Immortal having the definition of being able to survive the aging process and maintaining vigor and capability. Almost all of the "gods" of the Marvel Universe are "immortal until killed."
In the case of the Norse gods, called Asgardians, they are not quite immortal, they depend on the golden Apples of Idunn to prolong their lives. The Asgardians also understand they will face an ultimate reckoning called Ragnarok in which many of them (most of them) will indeed die.
There are many Avengers whose origins, powers or other circumstances grant them forms of longevity or immortality. This includes androids, mutations, cosmic beings, and aliens.
Immortal until killed
The Mighty Thor while not truly immortal, he like most Asgardians, will have a very long lifespan that requires an infusion of the magical essence of the Apples of Idunn. In the Marvel Universe, Thor has faced death several times during his career and for a time was forbidden to enter the realm of Hela, goddess of Death. He lived a life trapped within a suit of life sustaining armor because while he was undying, he was still subject to injury.
Hercules: a former member of the Avengers is a Greek god and by all accounts immortal and undying unless in a confrontation with other godlike beings. He and Thor have a rivalry that goes back hundreds of years.
Ares: Greek war god possesses the typical powers of an Olympian, including superhuman strength, speed, agility, durability, reflexes, and virtual immortality, though some of his powers are substantially greater than most other Olympians. Ares is functionally immortal in the sense that he is immune to the effects of aging and hasn't aged since reaching adulthood. He is also immune to any known terrestrial disease or infection. Since he exiled himself from Olympus, his durability has been reduced enough for him to be affected by conventional weapons.
Valkyrie has been a member of the Defenders and the Avengers. She is the strongest of all Valkyrior. Like all Asgardians, her tissue and bone are several times as dense as the mortal equivalent. Though she is not immortal, she ages far more slowly than humans. Valkyrie is immune to all terrestrial diseases and is resistant to most forms of injury. Her Asgardian physiology grants her superhuman levels of stamina. While her character has been transformed several times, the basic powers of an Asgardian were hers and she would have had the same lifespan as any member of Asgard as long as she consumed the Apples of Idunn.
The Incredible Hulk(s) - (She-Hulk, Red Hulk): while it has never been substantiated, the Hulk's regenerative abilities may mean he will never age or will age incredibly slowly in relationship to other humans. It is safe to assume all Hulks (red or green) may have the same nearly-immortal quality. It is unknown whether any of the Hulks will age while in their human form.
Firebird (Bonita Juarez) is a person who has been mutated by an unknown bombardment of radiation from an alien scientific experiment. The mutation has given her the power of pyrokinesis and precognition. She has also been proven to be unable to be killed by an Elder of the Universe. The source of said immortality is as yet unknown.
Mr. Immortal possesses the superhuman ability of immortality. Once he reaches the point of death, he regenerates from any and all injuries. When he wakes from his death, he is completely healed from any and all previous injuries and may wake in a rage. He also does not age. The cosmic being Deathurge indicated Mr. Immortal cannot be killed and will eventually outlive the human race.
Able to Be Repaired
Jocasta was a titanium alloy-based android whose mental capacities were designed and created by Hank Pym (Ant Man, Giant Man, Yellowjacket) and based on the engrams of his then wife, Janet Van Dyne (Wasp). Her android body was capable of surviving an incredible amount of damage and she is able to transfer her consciousness into other bodies based on her original design.
The Vision was a synthezoid created by Ultron (who was created earlier by Hank Pym) designed to infiltrate and destroy the Avengers. His synthetic body while appearing human was capable of a range of superhuman abilities including limited flight and density modification, making him able to pass through matter or become diamond hard and almost invulnerable. Despite these powers, the Vision was destroyed several times during his history, usually at the hands of incredible foes such as Count Nefaria.
Not Remotely Human
The Avenger, Simon Williams, codenamed Wonder Man, had his entire body replaced with a form of "ionic" energy. While he can appear to be a normal human, he has none of the requirements of a living being. He does not need to eat, sleep or excrete. He can exist under hostile conditions including space and does not appear to age at all.
The Sentry: Arguably one of the most powerful of the Avengers, has a host of superhuman abilities. Superhuman strength, speed and resilience, he is able to fight the entire team of the Avengers to a standstill. There are many theories around his powers and their origin. It has been suggested that he is not even human but an avatar of a near-divine being or force similar to Death or the Phoenix Force.
Jack of Hearts, whose mutagenic powers came from the combination of an alien parent and an immersion in the mysterious Zero Fluid, allowing him to project incredible amounts of energy. He has superhuman stamina and resistance to injury. He has restarted the energies of a dying sun at one point in his career. Having died more than once, his connection to the zero-point energy has revived him. What his body is composed of now is unknown but it is certain he is no longer human in the classic sense.
Beyond Human Longevity
Captain America has been in the peak of physical health since 1945. It is suspected the super-soldier serum protected him during his period underwater frozen in the Artic and has slowed his aging process significantly.
Starfox is a member of the long-lived offshoot of humanity known as the Eternals (as is his brother, Thanos). His body has been enhanced by cosmic energy to the point that it ages far more slowly than most humanoids and is superhumanly strong and resistant to harm and immune to terrestrial diseases. He can withstand impacts, such as falling from several stories or being repeatedly struck with superhuman force, that would severely injure or kill an ordinary human being with little to no injury to himself. However, he is far from invulnerable and can be injured by weapons, such as bullets or knives, composed of conventional materials.
Wolverine: a mutant that has been seen so far to be effectively immortal and has already lived a life significantly longer than a normal human. His body's regenerative processes make it likely he will continue to exist for decades to come. What such a long lifespan does to his mind, however, is another question.
As for being the final Avenger, well, the Avengers have grown since their initial start up. As such, he may be the final one of the original make-up depending on how Captain America ages.
In the one shot comic HULK: The End Hulk is the only person left in the world. It's just him and cockroaches. So really Hulk is the only immortal on the team.
Where does it say "gods" in the Marvel universe are immortal?
First, they aren't necessarily gods in the proper sense --- more like Clarke's "sufficiently advanced technology." Second, Thor, as originally conceived, wasn't the actual god, but just some dork who found the hammer and was granted the powers of Thor. Finally, if Thor is taken to be the actual Norse god as he is in more modern stories, then he is not immortal. Whether or not he is the last Avenger standing will depend on when Ragnarok happens.