In Tangled, after Mother Gothel stabs Eugene, Rapunzel wants to save him by healing his wound with her magic hair. However, before she tries to heal him, Eugene says:
Why does he say this? Why would she die?
Because she has made the bargain with Mother Gothel to never leave.
The film starts with the song When will my life begin hinting that up until the moment she escapes she hasn't lived, merely existed.
To abandon her freedom and remain with Mother Gothel implies she will no longer truly live, she will merely exist and die. Eugene knows that Mother Gothel will simply use Rapunzel until there is nothing of her left.
Of course, this is a bit of a complex reason, saying she will die is also an easier way of enforcing the weight of the decision she is about to make to the younger audience.
She won't die immediately. But the movie up to this point established that the hair could stop people (i.e. the witch) from aging and basically grant immortality and eternal youth.
So he means that she now has to die like every other human, since the hair seems to be the source of that magic and cutting it/using it up will presumably destroy said source.