I don't know about for phones/tablets.

The only genuine performance increase I could see is if you're using it for drawing/notes because the Apple Pencil samples at 240/s so you would see a difference in drawing smoothness between 60/120/240hz screens. This would mirror the use case in fast-paced and high action games - more info for your eyes sooner allowing better reaction times.

Also, isn't ProMotion just branded VRR? Allowing for lower battery use in static/low change applications?