I am a lawful permanent resident of the United States and have a study or work permit. Do I need to apply for an eTA?

No. As of April 26, 2022, lawful permanent residents of the U.S. must show these documents for all methods of travel to Canada:

a valid passport from their country of nationality (or an equivalent acceptable travel document) and
a valid green card (or equivalent valid proof of status in the United States)

Complete list of acceptable status documents for lawful permanent residents of the U.S.

You need an official proof of status as a lawful permanent resident of the U.S., such as one of the following:

valid permanent resident card (Form I-551)
foreign passport with an unexpired temporary I-551 stamp (also known as an Alien Documentation, Identification and Telecommunication [ADIT] stamp)
foreign passport with a temporary I-551 printed notation (“Upon endorsement serves as temporary I-551 evidencing permanent residence for 1 year”) on a machine-readable immigrant visa upon endorsement with a U.S. Customs and Border Protection admission stamp
expired permanent resident card (Form I-551) with Form I-797 (Notice of Action) for pending Form I-751 (Petition to Remove Conditions on Residence) or Form I-829 (Petition by Investor to Remove Conditions on Permanent Resident Status)
expired permanent resident card (Form I-551) with Form I-797 (Notice of Action) for pending Form I-90 (Application to Replace Permanent Resident Card [green card])
valid re-entry permit (Form I-327)
Form I-94 with an unexpired temporary I-551 stamp (ADIT stamp) and a passport-style photo