Prior to Muppets Treasure Island, it was normal practice for musical numbers to be performed in the studio and lip-synched on set.
Tim Curry recorded his parts in the studio prior to filming, but the energy in the performance on the day was so much more dynamic that the production team used the set filming audio instead of the studio performance for the movie, and so virtually all of the musical numbers are live recordings.
Thereafter the Muppets crew would try to use the live performances where possible for the more vivid energy on display.
Additionally, after filming wrapped on Muppets Treasure Island, Tim Curry was gifted a Muppet version of his character.