Is it ok to do "tribute" pieces in the honor of burlesque performers who are still actively performing, and if so, do you think it is best to ask for permission, or can you just do it?
Of course it depends. But generally, most performers would rather be hired than honored.
Sherry Britton worked with me on my tribute act to her. She worked on my costume, hair, music, and movement with me.
I think there are new performers who believe that many of us are doing acts out of a canon of burlesque acts, but most of us are doing our own numbers. Most of us would rather be paid than imitated; most of us would rather have the audience respond directly to us than to someone else doing a tribute to us. And even though many of the legends of burlesque are retired, their acts still belong to them, and it can be incredibly insulting to do a "tribute" without talking to them about your intentions before you begin to put the act together.
Recreating someone else's act is likely to be pure and simple copyright infringement. The US copyright office has a category that applies: http://www.copyright.gov/fls/fl119.html. An interesting example of this was when the producers at Burlesque at the Beach in Coney Island started, in the 1990s, to close most of the burlesque shows with a wine bath in a salute to Tirza, and called it "Tirza's Wine Bath." Tirza heard about it and sued to them to cease and desist. You can check out Tirza in one of my favorite books, Girl Show: http://tinyurl.com/tirzawb