Basically, there are two elements to the rolling resistance of a wheel on pavement. One is friction, which is directly proportional to the downward pressure of the tire at the contact patch multiplied by the area of the contact patch. Actually, it's a bit more complicated because pressure is higher at the center of the patch than at the edges, but the point is clear. The more force on the patch and area of the patch, the more resistance. So given that rider and bike weight are constant, you can reduce the contact patch by using a narrower tire and increasing air pressure, which increases pressure over the pavement, or you can reduce pressure over the pavement by increasing the contact patch (using a wider tire) and reducing air pressure.
The other element is deflection by the texture of the road surface. At high pressure deflection pushes against the forward momentum of the wheel. At lower pressure deflection is absorbed and the tire rolls more easily over surface imperfections.
So what does this mean?. First, the tire should be selected based on the rider's weight and the road surface. Heavier riders on rough roads roll better on fatter tires. Second, regardless of surface or rider's weight, a tire with a supple casing should roll faster. Since the tube is part of the package, using tubeless tires or using thin, supple latex tubes helps. Third, the relation between pressure and surface area favors surface area. In general, using a slightly fatter tire to reduce pressure more than compensates for the increase in the area of the contact patch.
There are no specific universal answers. In general, though, supple tires with smooth treads made from fast compounds, probably sized wider than you think you'd need, will probably help you go faster.