Dying Burning Bush

We're assuming you're worried about the die back on the burning bush but not the shrub as a pest, yes? (This shrub is classified as a pest because it self-seeds and chokes out native plants. It is now illegal to sell in Massachusetts for this reason.) So we'll address the die back here:
While it is impossible to tell from a photograph what has caused a normally hardy plant to have die back as we see in this picture, some of the possibilities are as follows.
1. Too dry. This seems unlikely for your plant as the plants that are on the ground are still green, but it's something that might be a part of the problem.
2. Physical contact with something that killed the leaves such as hot water from a sun-heated hose, drift from herbicide sprayed on other areas, cleaning products such as deck or house washing, liquid fertilizer that's too strong and sprayed on leaves, other garden products that made the plant sensitive to sunburn.
3. Physical injury such as damage from a string trimmer that cut the bark.
4. Borer that got into the main stem - check for holes.
5. Vascular disease - root rot fungi, bacteria etc. This plant isn't prone to such diseases but it is a possibility.

Sometimes a combination of one or more of these will be what does a plant in. If the plant is mostly dead, take it out and use this as an opportunity to plant something more interesting than a burning bush. The Blue Shadow Fothergilla, for example, has white flowers in early spring, bluish leaves all summer and fantastic fall color!