In the anime, it is shown that over usage of Byakugan and Sharingan eyes will cause the user to strain. Does that happen with a Rinnegan user?
They never mention strain of the Rinnegan in the manga so the following is conjecture
Nagato (and his puppets) seems to have the Rinnegan up 24/7. In flashbacks to the Sage of Six Paths he has Rinnegan up so I believe it does not cause strain like the other ocular powers
All eye techniques use up a lot of chakra.
Clan users can use the techniques for longer periods of time as compared to non clan users (like Kakashi). But even then there is a limit to which the doujutsu can be used, and that is dependent on the chakra level of the user.
Now to which extent the Rinnegan can be used has not been explicitly defined in the manga, so it can only be a guess. But surely it should cause strain to the user on over usage.