Acceleration on flat terrain is inhibited by various manifestations of inertia. One form of inertia to overcome is rotational inertia. A wheel at rest requires energy to be spun up to a given speed. The amount of energy required is affected by the rotational mass. A skinnier tire will be lighter weight, all other things being equal. A lighter weight tire will have lower rotational inertia, and therefore will require less energy to accelerate.
On an upward incline, part of your power output goes toward overcoming gravity. If you are on a 10% grade, you are lifting yourself and your bike 10 feet for every 100 of travel, or 1 foot for every ten of travel. A lighter weight tire will reduce by a few grams the weight you have to lift against gravity. Doesn't matter much for 100 feet of 10% grade. Matters a little more for 1000 feet of 10% grade, and possibly even more for 10000 feet of 10% grade. In the case of 100 feet, you're lifting yourself and your bike ten feet. In the case of 10000 feet on a 10% grade, you're lifting your bike 1000 feet. Weight starts to matter.
At 25mph, aerodynamics matter a lot more than at 10mph. So if you find yourself on flat terrain or on descents, a skinnier tire will have less aerodynamic drag than a fat one.
But all of these factors pale in comparison to the weight of the rider, the aerodynamic drag of the rider, and the total rotational mass of the tires, tubes, AND wheelset. A few watts savings can be had going skinny. Not a lot.
On the other hand, at 100 PSI you will have lower rolling resistance with a 30mm tire than with a 23mm tire at the same pressure. But people extol the ride-quality virtues of wider tires, and those come into effect if you drop the pressure a little, in which case rolling resistance might increase a little.
But wait, there's more; a wider tire at a sane pressure will be less sensitive to the bumps and vibrations of the road, and may maintain better ability to roll with the punches. That can improve performance a little.
What does it all mean out on the road?
I've ridden 23, 25, 28, and 32mm tires. I currently have 28mm GP5000 on my road bike and 32mm GP4Season on my hybrid. I'm slower on my hybrid, but it's not because of the tires. It's because my riding position on the hybrid is more like a sail catching the wind. And because the hybrid is ten pounds heavier with rack. But even then, I often loan my hybrid to my brother so he can ride along with me when I take my road bike on rides in the 16-28 mile range. He keeps up with me just fine except on descents where it's the aerodynamics that give me an edge. On flat terrain I'm just a little faster, and on 10% inclines he's right there with me.
Look more at the tire's performance characteristics, and less at its width. 30mm is fine. I'm perfectly happy riding 32mm tires on the hybrid. What I hate on-road is tires that are knobby or have a tread that is not slick where it contacts the road under normal riding. That sound of buzzing tires with tread make?.... that's the sound of watts being stolen from you.