Originally Posted by godspiral

According to this calculator, http://www.kreuzotter.de/english/espeed.htm :

going from 1.75" to "narrow racing tire (say 21mm. 1/2 the knobby tire) will cut the overall air drag 20%, and cut 8x the area difference (700 x 21mm) of the tires.

Going from 23mm to 20mm, you are cutting 4% air drag in the flatbar position. Probably 6% in drops.

Many tire manufacturers have found that thinner tires at the same pressure increase rolling resistance (RR). So thinner and lower pressure would likely increase RR.
Even if narrow tires increase RR, they can be faster if you are going fast. (Air drag slows you down proportional to speed squared, wheras RR is only linear with speed)