At same pressure, riding on hard surface like pavement, thinner narrower tires are:
- Faster due to less air resistance than wider tires
- Faster due to less weight, mainly on climbs, but do note that weight itself also increases rolling resistance
- Slower due to more rolling resistance than wide tires, IF they are run at the same pressure
So it's complicated. Also, nobody runs narrow and wide tires at the same pressure. Firstly, the casing tension is proportional to tire width and tire pressure. So the wider tire you have, if you run the same pressure, the more casing tension you have. Eventually the casing tension is so high that the tire can't withstand it and explodes.
Also, at the same pressure, narrower tires have a long and narrow contact patch. Because the contact patch is long, it's deep. Wide tires have a short and wide contact patch. Because the contact patch is short, it's shallow. These two contact patches have the same surface are but different shapes. So at the same pressure, the wide tire deforms less vertically. Thus, it's less comfortable. This too explains why narrow and wide tires aren't run at the same pressure.
Also narrow tires are often run at very high pressures due to possible pinch flats. Wider tires are less susceptible to this, so they can be run at very low pressures.
So there are three factors to consider when deciding tire pressure: casing tension, how much the tire deforms vertically, and whether the tire at this pressure has enough resistance against pinch flats.
The question of whether wider tires have more rolling resistance than narrow tires is therefore very hard to answer, since you have to have some law of how to vary the tire pressure as a function of tire width.
However, all of this changes on loose surfaces. You can test this as follows: put 28mm slick tires on a road bike, pump them up to 7 bar, go riding on soft sand road that is wet (this is best done in regions that have snow because the melting water from snow will make the road deeply saturated in water). You will immediately notice that you will slow down quite a lot compared to hard pavement or even the same sand road when it is fully dry.
On loose surfaces, it is heavily beneficial to have a wide tire at low pressure instead of a narrow tire at high pressure. This is so beneficial that on loose surfaces, it almost always overcomes any air resistance drawback from the wide tires. If you can't ride more than 20 km/h due to rolling resistance slowing you down, air resistance is anyway so small that it doesn't matter. Reducing that rolling resistance matters a lot.
Because rolling resistance is a constant force and linear power, and wind resistance is a square force and cubic power, whether or not air resistance or rolling resistance matters more depends on the speed. On hard surfaces, it's hard to answer. On loose surfaces, the drawback from high rolling resistance of narrow high-pressure tires is so great that less rolling resistance almost always wins.