Rented Property: Install Nest
Assuming the Nest thermostat you want is compatible with your boiler, there is no issue in replacing the existing one. Most if not all thermostats are designed to be removed in a non-destructive manner.

Mounting it may require drilling new holes. When your lease is up, simply reattach the old one. Any new holes if needed would be covered up by the thermostat, hence should not be held against you by any reasonable person.