Do you have to be a priest or nun to become an exorcist? Also, do eastern rite Catholics perform exorcisms?

You must be a bishop or a priest who a bishop delegates. Yes, they do.

An exorcist is a bishop or a priest appointed by a bishop, who has a special permission to perform exorcisms. The permission is permanently or temporarily granted by the bishop to priests.

Yes, Eastern Catholics have exorcists too.

Canon Law of the Catholic Church # 1172 states:

Can. 1172 §1. No one can perform exorcisms legitimately upon the possessed unless he has obtained special and express permission from the local ordinary.
§2. The local ordinary is to give this permission only to a presbyter who has piety, knowledge, prudence, and integrity of life.

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