I have never taken off the burlap and basket when planting. I haven't had a tree or shrub die from it but i have had to replace them when the homeowners didn't give it enough water or over fertilized.
I was taught in college that there is no need to remove it since the roots will grow right through it. I have been to many nurseries where the roots are growing like crazy through the burlap.
The reason they died are more likely from some other reason.