Boy, burning bushes have massive root systems that are very difficult to dig out, especially if it is an established plant. I had to dig mine out from where it was because it had been planted too close to the foundation and the roots were trying to burrow through. It took years to get all the roots out and we had to pull the plant out with a truck.
You can move it because they are hardy and will adapt once set in but it is very difficult getting them out. You need to dig very deep around the out side of the root system. Once that is done, you will need to pull it out the best way you can. We used a truck and for four after we pulled it out, i was still digging out roots.
Good luck and if you can affored it, you might want to get a landscaper to do it for you.