Good afternoon and thank you for reaching out to ask2.extension.

 It has been a tough year for lots of plants.  I believe it was mostly due to the harsh winter, tons of snow and the cold, wet spring.

Have you found any bugs on the leaves and/or the branches?
Dying limbs you show in the picture could be Mites or Scale.   Spider mites are fairly common with burning bush. If spider mites are present discoloration, browning leaves, dead wood, and leaf drop will all occur. To check for spider mites hold a white sheet of paper under the foliage where it's brown and green. Tap the branches. If you see little specks moving around that are about the size of a pencil tip, either orange to red in color or greenish with two black spots, then your plant has spider mites. You can use a systemic insecticide or spray with Neem Oil.

Scale is an insect that attaches itself to the backside of leaves or sometimes on the branches.  It looks rough.  This is another insect that burning bushes are troubled by.  Spray twigs/branches with a  horticultural oil or insecticidal soap.  This is usually most effective early.

I don’t know how dry your area has been.  One good preventative for both pests is making sure the shrub has enough water during dry periods.

Keep an eye on the shrub.  Depending on how far gone you plant is, you may lose the plant.  Watch for anything happening to plants/shrubs that are near.