AKA Diamond, Gordon Theater - Demolished in the 1980's.
According to Joe Vogel on Cinema Treasures, the Lake was on 4th Ave, a few doors north of 2nd Street. I visited the town on June 22, 2022 and that seems right to me. There is a building that can be seen in the old post card views a few doors down from that location. That would put the theater at the location of the Lake Odessa Community Library (roughly) which is newer construction. I will use that address for reference. See the photo analysis below for more information.
I also found articles on the former owners/operators. The more recent one was Charles D. Gordon. He bought the Lake Theatre and a candy shop. He re-named the theater to the "Gordon Theater" and featured live entertainment at that point. He later died tragically in a car accident. Prior to that was the Chase family. Wayne Chase was the son of Hollis who ran the theater as the Lake prior to that. Hollis Chase owned it from 1924 on until he died and Wayne took over. At some point after the 50's or 60's the theater was demolished.