In Naruto, does the previous host die after Orochimaru changes hosts? If so, why are there some who are eager to be the next "container"?


  And in Boruto why haven't they stopped Orochimaru from taking new hosts?
To answer your first question,Yes. Much like shedding skin of a snake. 

They haven't stopped him in Boruto because he is much more useful to them alive. 

They know Orochimaru has huge amount of Intel, which would be useful to them in the future. This proved to be true in the Shin Uchiha arc, when he told them who Shin was.

Also, they know that now that Naruto and Sasuke are so much powerful (too OP), Orochimaru wouldn't be a big problem to them.

But to prevent any mishap, they still have put surveillance on him. Captain Yamato was seen keeping an eye on Orochimaru in the Shin Uchiha arc in the anime.