It is not possible to determine why the shrub failed to leaf out from a photo. Pruning this spring may have been a contributing factor in its failure to leaf out but it is unlikely it is the cause of it failing to leaf out. The roots can be checked by excavating gently at the base of the shrub to see if they are healthy. If they are healthy the shrub may leaf out this year.
There has been a lot of rain this year, if the shrub is in a low spot and stayed wet too long it may have drowned.

Thank you for the speedy reply. What would healthy roots look like? Is there a way to tell if it drowned? One source suggested cutting the burning bush way back to a couple of feet - like we have done with lilacs in the past. What course of action do you recommend? Is there any chance it will come back next year if we do nothing and it never does leaf out this year? I think you implied it might actually still leaf out this year? Thanks.

Deb

Thank you for the speedy reply. What would healthy roots look like? Is there a way to tell if it drowned? One source suggested cutting the burning bush way back to a couple of feet - like we have done with lilacs in the past. What course of action do you recommend? Is there any chance it will come back next year if we do nothing and it never does leaf out this year? I think you implied it might actually still leaf out this year? Thanks.

Deb

Healthy roots are white and firm. See the images in the fact sheet for what healthy roots look like.
http://www.missouribotanicalgarden.org/gardens-gardening/your-garden/help-for-the-home-gardener/advi...

I would not cut back the bush unless you are sure the branches are dead. Nothing will change if it is dead and if it is alive it may leaf out.