No. As per dictionaries, Dhan te nan has no meaning.
Well, Dhan te nan (or tan te nan) is a very Desi representation of a typical old espionage tune, and something I could never expect to reach such a level where Gulzar made it go. But then, Gulzar is a person who gets the most earthy lyrics and so is dhan te nan.
If you have seen those '70s movies of Amitabh Bachchan n all (like Don, for instance) where the hero is running away n villain is following or vice versa but both are in different frames at different times, there was this old espionage music that went like tan-te-nan, something like the music of the song. You can also see many people say the word in many movies just for an effect. One place I remember is Rajesh Khanna saying the word in Bawarchi when he brings tea to the old man of the family. There he says it in his own style, something like taan ta raan.
So, that's the history of the word I could get. But if you're looking for a dictionary meaning, the answer is, it doesn't exist.