A bijuu can indeed die, in a way.
A bijuu is purely made up of chakra, some times the are even referenced to as Chakra Monsters. The same rules apply to the bijuu as to all the normal humans in the world. If your chakra runs out, you die.
quoted in naruto episode 50-60, or when ever it was that the chunin exams went on and the sand attacked the leaf, sasuke and kakashi were off training when kakashi told sasuke "well your about good for 2 blast", refering to chidori. then sasuke said, "what would happen if i tried a third?" kakashi then said "in the turn that you try to force a jutsu, not only will it not work, you use up all your chakra, worst case scenerio, you die" so if you do lose all your chakra, you do die. as quoted from
This means that if a bijuu runs out of chakra he dies. If the jinchuuriki dies with the bijuu still inside of him the bijuu will die.
in this last case there has also been stated that if the jinchuuriki die's with the bijuu inside of him he we revive
in chapter 503. On page 14 Kushina tells Minato her plan to reseal the Kyuubi within herself and then commit suicide. That was she says that the Kyuubi's resurrection will be delayed.
so upon the death of the jinchuuriki the tailed beast will disappear for a while. But it'll reappear again after some time. It is also said that this reappearance is on a random location. If the bijuu retains his original memory is also not known.