Immure literally means “to wall in”— i.e., to place someone inside a wall; however, it is used metaphorically, in many useful ways:
The archaeologists found precious artifacts immured in the walls of the tomb.
The students immured themselves among stacks of books before exams.
Inure means “to harden” (a person)—i.e. to make a person insensitive to difficult conditions:
After years of living in New Orleans, I am inured to the heat; it does not bother me.
Both words are used mainly as transitive verbs, but in the passive voice, as shown in the above sentence examples. In other words, although one can immure someone with something or inure someone to something, we normally say that people are immured by something, and inured to something.