What visa should an Australian apply for to live and practice at a Buddhist monastery in Thailand?

Perhaps DTV Destination Thailand Visa?

You will require the following steps to be completed; It is mandatory that the following steps are taken before the Non-Immigrant “ED” visa for the Monk can be issued:

1. Recommendation letter from the temple in Thailand sent to the Office of Buddhism, Thailand.

2. Recommendation letter from the Office of Buddhism, Thailand, sent to the Ministry of Foreign Affairs, Thailand.

3. Approval letter from the Ministry of Foreign Affairs, Thailand, sent to the Royal Thai Embassy.

4. The Royal Thai Embassy issues Non-Immigrant “ED” visa for the Monk (visa applicant)

You should apply for a Non-immigrant Education visa to study as a monk.