The fastest version is the standard US version, PRG0 or PRG1. The JP version has longer loading times for each level, making it a lot slower. And PAL runs at a lower framerate and is generally bad.
There is some general information about the game found here:
http://tasvideos.org/GameResources/NES/SuperMarioBros3.html
Though some of the things are not up to date and a lot of info is lacking. In general, there isn't really a lot of explanations on how to TAS it optimally...
I've been TASing the game for many years now, and still does on a daily basis. I guess I could produce some kind of TASing guide for the game, if I'm not lazy enough... I could however assist in various questions and such. A good place to start is to watch the current TASes of the game and try to replicate what you see to get a feel of it.