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The following article should be of some help in treating the rust: https://www.gardeningknowhow.com/plant-problems/disease/learn-about-plant-rust-disease-and-rust-treatment.htm

This article will help with pruning: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

This article will help determine if the bush is still alove or not: https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm

Normally, you should wait to hard prune in spring to protect the plant over winter from receiving any damage. If the plant is still alive, you can likely trim back some of the dead growth but I would be sure to leave some for winter and then prune it back by a third prior to its new spring growth. This article will help: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

You may also want to up its water, if possible, especially if you continue to have these high temps.

Hopefully this article will answer your question: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

This article should help you with that: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

I know that you do not think it is a pest or disease, but given that it happens only t the burning bushes, it is likely a euonymus specific disease or pest, such as euonymus scale. Can you please describe the pattern of the dieback on the plants? Do they suddenly wilt and die or is there damage to the leaves first before they die? If you can let us know that, we can better identify what it might be.

Although burning bushes prefer acidic soils to alkaline ones, they are not acid loving plants. They prefer moderately acidic soils with pH levels between 6.0-7.0. As for fertilizer, you can use a general, balanced type or one that is aimed towards acid-loving plants like azaleas and rhododendrons. For additional inofrmation on caring for these shrubs, this article will help: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm