Too many people simply repeat the short version of what they've read or heard, they leave out some important details.
I live in a place where burns and extensive tree harvesting occurs and I can tell you that, in regards to elk, there is no canned answer to the OP's question. There are several things at play and the most important is that elk are browsers, not grazers, they have a pretty varied diet but they have preferences for certain foods. While grasses are important to their diet, they prefer specific grasses. At the same time, if there is no grass then they will eat the fresh tips of young trees and brush. They love forbs (flowering plants) but if they can't find them then they'll move to another food.
When a burn occurs, it can be so intense that it burns the nutrients out of the soil, in which case it takes a lot longer for plant grown to start again. Plants grow through a cycle and the types of browse that an elk will eat is mostly toward the end of that cycle. Grasses will start earlier in that cycle but, as I said, elk prefer specific grasses (bunch grass, wheat grass) and those are not the first of the grasses to regrow. Also, it takes water to make grasses and trees grow and if your hunting area doesn't get a lot of water then the plants will not grow back very quickly. Finally, elk don't mind going in to an open area to eat, but they usually prefer to be in a grassy area with trees close by, enough trees to provide cover, and they typically prefer pretty dense tree cover, just a handful of trees in the middle of a large burn wont work.
In my opinion, over the years, the idea that a burn makes a good hunting area of one of the old wives' tales. It used to be common for a farmer to burn off his fields so that he could eliminate weeds and other unwanted plants. Because this was farm land and the farmer then replanted after the burn, and there was regular rains, the new plants would grow pretty quickly. As a result, a lot of people have jumped to the conclusion that the same thing always happens in the forests...it doesn't.
We've had a lot rain this year where I live (rural, southwest Montana) and I've got grass and trees sprouting in places that there hasn't been any growth in for the last seven years. We have burns and extensive tree cuts that haven't had any reasonable new growth for at least the last three years that are now showing new plants everywhere. But extensive rains have made the difference and I'm sure that we will see it effect where the elk move to.