Which needle size will give me the longer scarf?
Take a yarn in a similar weight, measure out a few yards - knit it with the smaller needles you considered (3mm).
Take that same yarn and measure out the same length - knit it with the larger needles you considered (6mm).

Measure both.

I really think the trade off between "more yarn used in a stitch but taller stitches" on the larger needles and "less yarn used in a stitch but shorter stitches" will be neigh-on impossible to predict without either a swatch or at least physically measuring a stitch.

My prediction is that the larger needles will result in a longer fabric for the same length. This is based on the idea that larger needles = a less dense fabric which means more "air space" incorporated. A smaller needle will result in a denser fabric which means more space filled by yarn and less by air.
This is a theory, please at least swatch in a different yarn.

This may also help. I actually just stumbled on it yesterday:

http://www.knitdarling.com/blog/needle-material-affects-on-gauge

The link contains a summary of an experiment a blogger did on how needle MATERIAL affects gauge. The metal needles made the longest test piece for her and the resin needles made the shortest.