That is a good and interesting question.
It is a beautiful plant that unfortunately is very invasive and escapes into wild areas, displacing native species.
We are glad that you are aware of this.
We are not sure of the answer to mitigate their invasive qualities- and I'd assume if there was a way to do it we would be aware of it.  I do know from personal experience that you can cut those huge shrubs back to the ground and they will still sprout, grow and recover. 
I 'removed' three huge bushes by cutting them down to stumps and treating the cuts with herbicide and I am still getting some new regrowth years later (and sprouting seedlings that thankfully are easy to pull.) 
https://extension.umd.edu/resource/burning-bush/
I'm not sure if they put out flowers/berries on that newest growth- if they did you could probably shear them off.
But that is work in and of itself- imagine the beauty of replacing those shrubs with native blooming or berrying shrubs like Highbush Blueberry (the kind we eat and love) Red Chokeberry, some Viburnums,  Va. Sweetspire, Ninebark and so on. I replaced the large area where my 10x10 foot burning bushes were growing with a pollinator garden. You can learn more about that here:
https://extension.umd.edu/resource/pollinator-gardens/

I understand that they number that you have makes it feel overwhelming. Maybe control just a few at a time. Anything is a start in a better direction.


Christine

Thank you for your response. 

Will herbicide applied to the stumps get into the soil and contaminate / harm other plants in that same area? 

It depends on which herbicide you use but most systemic herbicides stay within the plant system. Glyphosate binds to the soil and eventually breaks down so it will not travel to other plants. You can paint it on the fresh cuts so the spray isn't drifting to other plants. 

Some of the other systemic herbicides can move through water so travel more in the environment. Dicamba is an herbicide that could be harmful to other trees, shrubs and perennials since they can absorb it through the roots. 

You may have to paint on the herbicide and remove shoots multiple times to get it from resprouting. Burning Bush are tenacious. I removed some about 10 years ago and still find seedlings pop up once in awhile. I replaced with the Red Chokeberry that Christine mentioned above. Their red berries persist through the winter more and are more noticible than Burning Bush. They still have great fall color too!

Let us know if you have further questions. 

Emily