It is about 14 years old. The other one we planted died about 2 years ago.
Burning bush spreads by underground runners and by seed. It is considered invasive in some areas. Though it's not often bothered by insects and disease, it can happen. This article should help:
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-turning-brown.htm
I planted a Burning Bush from a pot about 10 years ago. But during the summer months the leaves never turn dark green. The Fall season brings the brilliant Red leaves the plant derives its name from but I'm concerned it may not be healthy.
Unfortunately, your photo did not come through. Depending on the exact cultivar, some have leaves that remain colored, or even on the yellow side.
Otherwise, you will need to test your soil for pH imbalances or deficiencies. Some plants will exhibit red coloration with phosphorus deficiency, or acidic soil. In the case of this shrub... That is not a good way to diagnose deficiencies, since it will be colored to a degree, already.
Here are some articles that will help:
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-ph-plants.htm
There are a few green leaves coming but for the most part, they look dead.
I'm concerned you pruned them too late in the year and they didn't have time to harden off before the freezes started. They normally should be pruned in late winter or early spring before the new growth starts.
Do the branches bend or do they snap off? If they snap off they are likely dead. If they are bendy, they may still recover.
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm
I want to make a hedge. How far apart do I plant the original plants?
Plant spacing is determined by the plant species and hedge type. Plants in a sheared, formal hedge are generally planted 1 to 2 1/2 feet apart. Shrubs in an informal, natural hedge should be spaced further apart. A spacing of 2 to 4 feet is appropriate for medium-sized (6 to 8 feet tall) shrubs. Large shrubs (8 to 12 feet tall) can be spaced 4 to 6 feet apart. Check the mature sizing on the variety of Burning Bush that you choose.
I just discovered this on my shrub. I have two burning bushes. The other one is fine at the moment. What is this and how do I treat it?
They are one of the many types of Aphid! Here is an article to help you control them:
https://www.gardeningknowhow.com/plant-problems/pests/insects/homemade-aphid-control.htm
Can you trim a burning bush after it leafs out, in June?
If you want to do light pruning for shape you can do it now. If you need to do extensive pruning or rejuvenation pruning (cutting it 6 to 12 inches from the ground), it's best to wait till late winter or early spring while the shrub is still dormant.
Looks dead, just a few leaves like a dozen, do I leave it and hope for next year or get rid of it?
Burning bushes are hardy to zone 4 (you are 4b) but perhaps you had an extra harsh winter. You can break off a couple inches of one of the stems or just scrape off some bark. If there is green showing, then it is still alive. If the inside is brittle and white, I'd say replace it.
Here is more information on burning bushes:
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm