Thor would likely die or at least be severely hurt, and they are not immortal. The idea that they are immortal comes from human Norse mythology.
- Loki's father the ice king was killed by Thor's father Odin.
- Odin must go into the Odinsleep else he would die, and while in the sleep he is mortal.
- Thor will inherit the thrown of Asgard when his father dies.
- Oden (Thor's father) is the son of Bor and Bestla who are now died.
Why was Thor afraid of being trapped in the cage, and why did Nick Fury put Loki in the same cage? It's because the Hulk is the only known superhero to best an Asgardian. There are several comic books, television shows and even in the Avengers films where the Hulk beats up an Asgardian.
In one Marvel animated film Loki uses mind control over the Hulk, and uses him to attack Asgard. Where the Hulk destroys most of the city before Thor is able to break the mind control spell that Loki was using, but at one point in the film Thor tells another Asgardian that he expects to die when he faces the Hulk.
In the Marvel animated film Planet Hulk. The Hulk is outcast to another planet where he battles Beta Ray Bill. Beta Ray Bill is the only other character to ever receive his own Asgardian hammer of the Gods. The same weapon used by Thor. Beta Ray Bill is the only character to match Thor's strength and good will. Still, Hulk beats Beta Ray Bill to an inch of his life before he finally stops.
In the Avengers film Hulk beats up Thor, and would have won had a fighter jet not distracted him.
Loki traps Thor in the cage, because he does not know if Thor could get out. The cage can be ejected from the carrier, because Nick Fury knows it wouldn't hold the hulk. So it's not an indestructible cage. When the cage is ejected Thor struggles to get out, and at the last second manages to escape but is knocked out in the process.
Could it have killed him? Yes. Would it be guaranteed to kill him? No, but you if want to kill an Asgard. Send the Hulk after them.