Answer: Burning bush requires at least a half day of full sun and a well drained soil; it can suffer root damage in overly wet soil.
It is possible the excess rain created overly wet soil conditions, with that being indicated by the browned foliage. If this is the case, the foliage will probably drop and then be replaced -- if the plant is going to survive. There is not much you can do except wait and see and be patient. However, I just looked at the precipitation record for your zip code and it doesn't seem like that would be enough rain to cause that type of damage (about five inches over a week's time), especially if the plant has been in place during spring rains combined with snow melt, for example, with no ill effects. An exception might be if surface drainage conditions have changed and so excess runoff was directed at the plant causing saturated or flooded conditions with standing water.
When a sudden change occurs in a plant, it can be due to many different factors. A common one is accidental herbicide drift or overspray, another is physical damage to the trunk or roots (weedwhacker, voles), or an underground obstruction (a large rock) is preventing the roots from growing deeply. Sometimes this type of cause can take some detective work to discover.
In the meantime you could also check for signs of scale insect infestation or other pests, and you might also consult with your local county extension to see if they are aware of any particular problems affecting these shrubs in your local area recently. I hope this helps you trouble shoot.
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