Where is this from?
I’m not familiar with the phrase. How is it used? Can you give us an example?
I think it arose here on the boards with a poster who got angry and said “I burning your dog” - things being generally what they are here, dopers latched onto the flawed syntax, completely ignoring the actual content of the rant (same thing happened before with the Og thing and probably others too).
Here it is.
Never pet a burning dog.
Unless you’re at Burning Man.
Or have a burning desire!
That has to be, absolutely, the funniest thread I’ve ever read!
… and the next thing you know, I wiping out all your base with a 1920’s-style “Death Ray!” 
This post has been you-know-what by you-know whooooo!
Wow, look at all the great Dopers in that thread who seldom, if ever, post anymore. Memories, memories . . .
Another small portion of my ignorance successfully fought. Thanks.
It comes from the 1980’s song Rio, by Duran Duran.
Yes, thats right. “Her name is Rio and she burning your dog….”
But only once.
For 20 minutes. In 1960.