We've been taught from youth that dying one day is natural and inevitable, so obviously, any type of immortality is seen as against nature, so there must be a drawback.
But immortal being from birth, like Tolkien elves, for example, are immune to that kind of mindset. Tolkien never mentioned that they were bored of their eternal life. Neither are the Gods.
It's just in the head that you think it'd be boring.
Personally, I think eternal youth would be nifty (rather than immortality), as long as you can spend it with your loved ones.
In the reference frame of humanity multiplying like rabbits, yeah, there must be some drawbacks.