Among similar examples, it is common for Anglophone gays to use words that originally had more vitriolic connotations, e.g., 'queer,' sometimes 'dyke,' and to a lesser extent 'fag,' as communal identifiers. I am led to understand that a something similar has happened to 'Tunte,' though my more-fluent friends can't elucidate to what extent this is the case. In what situations would this word be used in present-day Germany?
There may be differences across various demographics, but in general, I think that the closest equivalent in English would be queen -- at least as far as meaning is concerned.
Wikipedia seems to think so, too:
Mit dem meist abschätzigen Begriff Tunte werden sowohl in manchen heterosexuellen als auch homosexuellen Kreisen oft solche Schwule bezeichnet, die durch ein affektiertes Verhalten auffallen. [...] Besonders in der Selbstbezeichnung unter Schwulen muss es jedoch nicht in jedem Fall negativ gemeint sein. German Wikipedia "Tunte"
In gay slang, queen is a term used to refer to flamboyant or effeminate gay man. The term can either be pejorative or celebrated as a type of self-identification.English Wikpedia "Queen"
Note, however, that there are subtle differences. "Tunte" originally denoted a dowdy kind of woman or an old spinster -- so a much less glamorous derivation than the English "Queen".
Usage is even more different. The German "Tunte" is much less affectionate and tends much stronger towards pejorative use than the English "Queen".
As always with slang: Do not use it, unless you're very sure it's appropriate in context. It's super-easy to cause unintended offence. Especially if you're straight, I would strongly advise against using "Tunte". The queen in question may well punch you in the face -- or at least make a very cutting remark indeed.
Long story short and to answer your actual question: There is no situation whatsoever where a straight non-native speaker can use "Tunte" safely. A gay non-native speaker may be cut some slack, but it's still risky.