Is there a specific name for a figure of speech that’s acted, for instance: “Thars gold in them thar hills!” stated in the voice of a grizzled prospector?
Not sure exactly what you mean. If you mean the use of “thar” instead of “there’s”, that’s called “dialect.”
I’m thinking more of a new form of English expression where the phrase is said as seen in the original medium like “We don’t need no stinking badges!” as both in context to the situation and as a reference to the movie The Treasure of the Sierra Madre. Often these figures of speech are more commonly known than the original instance, for example, I’m not sure what my first example is from except that it’s probably from a movie or TV. Obviously this mimicry or acted phrase is a fairly recent (impossible pre-radio, say) mode of communication. Just wondering if it has a name yet.
I don’t think it has a name, but I’d be interested in learning the nomenclature if there actually is any.
It’s hardly impossible that pre-radio folk imitated people of other lands and other classes saying things that they have heard from rumor; such mimicry would not be accurate, but it could become standardized, so to speak.
The only word I know of that’s close in meaing is cliche.
Catch-phrase, cliche, idiomatic expression, quotation. reference, etc. There are so many terms that fit, it’s unlikely that a new word will be created.
This is a bit of a hijack, but this being the Straight Dope and all, I feel the need to clear this up. “We don’t need no stinking badges” comes from Blazing Saddles. In The Treasure of the Sierra Madre, it went:
Continuing the hijack, “We don’t need no stinking badgers!” is from UHF. Glad I could be of service. 
Regarding the OP: I suppose it could be regarded either as an affectation if the person (or persons) did it for more than just the catch phrase. Or it could be argued to be a cinematic allusion.
In regard to the argued quotation, clearly it dates back to Treasure of the Sierra Madre even if the quotation was mildly changed. To give it credit to either the later Blazing Saddles or the even later UHF would be like giving credit for the phrase “Play it again, Sam” to Woody Allen in the play/movie Play It Again, Sam rather than Casablanca even though Rick never actually said the exact phrase.
“You know, Louie, this could be the start of a wonderful friendship”
Thanks. Clearly all later usages are direct references to The Treasure of the Sierra Madre regardless of phrasing.
The subtle difference between a pre-radio phrase and a post-radio phrase is that a pre-radio phrase would not be consistently acted or mimicked. Compare: “To be or not to be.” and “What we’ve got here is a failure to communicate.”
I’m not looking for a Great Debate or to add to the English language, I guess my question has been answered in the negative.
Uttering a statement “out of voice” without regard to the original source would generally be regarded as an affectation.
If the OP inquiry was about the concept of a statement carrying meaning from the original source then it might loosely be described as an allusion. That is, this situation is somewhat like the one in the movie everyone knows of.
Sure, it’s a reference, but it’s wrong to say it’s a quote. Rereading eunoia’s post, I realize s/he didn’t claim it was a quote. I’m just spreading trivial knowledge, since the mistake is commonly made.