Yes, there are resources that you want to prevent API users from modifying, but they are application dependent. For instance, an API I'm working on right now lets callers read but not update audit data, read user records (but only modify parts of their own), and create and update home addresses.
You will want to make sure that you have rigorous security in place to prevent users from modifying certain parts of a User (such as username or password), especially if user A is calling PUT /users/B
.