Unless necessary for safety, I would think they should go.
The University of Minnesota Golf Course lost a large percentage of its trees to Dutch Elm Disease a decade ago or so. After we got done mourning their loss, it quickly became apparent that, if anything, the course improved.
At my course, we got rid of a bunch on a par three last winter. The visual change was startling. Nonetheless, even though I was co-chairman of the green committee I did not hear a single negative comment about the change. The condition of our green improved dramatically and will get better with time.
My primary purpose in my role with my club is to chop trees and prevent dumb, permanent mistakes.