We burned ours without mowing since it had already laid over and all the seed had burned off. We sprayed it and killed everything 7 days prior. If it was still standing up straight with seed on we would have mowed then burned.
If it is still standing with seed and you have access to a tedder it's one of the best ways to do it
Doves love bare dirt, and burning "shells" seed. We've had outstanding dove hunts over accidentally fall burned native pasture. The birds love wild seed laying on the ground.
The other reason I've heard is that burning prevents/deters the seed from germinating after some wet weather making it viable and attractive for longer. The bare dirt thing is important too.