Is there a list of all monsters with Immortal nature, like the Nagas, because I can't find any. 
I want to know what kind of immortal creatures there are (other then elementals, fiends, constructs and undead), to know what I could use in a dungeon, the kind of "long forgotten ruin" where anything living would have died of stavration and all. 

What sort of immortality are you looking for? Immortal in the sense that dragons can live many thousands of years? or Immortal in the sense that they have been around since the beginning of time itself?

There are many immortal, or ancient races and monsters that have a variety of different themes and present a different atmosphere.

If you are looking for a list of monster manual creatures that are considered "immortals" 'Forgotten Realms Wiki' has the exhaustive list.

https://forgottenrealms.fandom.com/wiki/Category:Immortals

Dragons are not immortal in any sense. Immortal means forever, and a dragon’s lifespan is like... 2-3 elf lifespans, a significantly less significant difference than that between elves and almost any other race (I mention this only because elves are so commonly played; if every race that lived a long time were super niche, I’d agree that the relative lifespan is “functionally immortal” to, like, a human; but a dragon to an elf is like a halfling to a human, and everyone plays an elf, so I’m really not on board with dragons being “functionally immortal,” haha).

But OP is referring to a specific mechanical feature of nagas, “Immortal Nature,” which means they don’t need to eat, drink, sleep, or breathe. And this would be easy to do if D&D Beyond’s search bar were anything approaching functional, but it notoriously is not, so unfortunately your best bet is to just guess at what you think might be appropriate and look at the description.

Don’t be afraid to just add the trait to anything you want if you think the monster otherwise fits your dungeon thematically. The players shouldn’t be looking in the Monster Manual to check for “Immortal Nature” anyway, haha.

Well warforged don't count as a construct functionally.

But a little homebrew could be in order, such as they got cursed or even blessed, I think cleric, druid, or monk gets something like that, or something that really delays it. Still food for thought.

There are all sorts of monsters that you can easily justify as being able to survive for a thousand or ten thousand years without sustenance.  You can reason that certain oozes or slimes can, for instance, go dormant.   There are also semi-Elemental creatures that could either survive without food or which are capable of making their own, like Water Weirds, Mephits, or Djinn.  Some celestial creatures likewise do not need to eat or are likely assumed to not need food even if they can eat: archons, couatls, and the like. 

Nothing is immortal except Death itself. 

Kurgan has spoken.