Pruning is not my strong point. So I consulted my Ortho's All About Shrubs book.
Forsythia: Bloom in Feb., Mar., or Apr. depending on climate. Prune annually AFTER flowering by removing 1/3 of the oldest canes. DO NOT SHEAR. Renew old plants by cutting them to the ground.
Burning Bush: Pruning destroys its neat outline and causes uneven growth.
Ortho's All About pruning, Yews: The time to shear is in winter or early spring so that the new growth will hide the cuts. Then lightly shear again in early to mid to midsummer if new growth warants it. DO NOT wait to long to shear. Shearing an evergreen in late summer or early fall leads to soft new grow that may be killed by cold weather. The snipped leaves will be unsightly all winter long.
Like I said shrubs and pruning are not my strong point. That is why I buy books to have a refrances to look at. Once I finish my Turfgrass class through PLCAA. I'm going to take their ornimental class. Educating yourself is the best thing you can do.
You can prune forsythia anytime but they flower on last years growth so the "correct" time as with other early spring flowering bushes is soon after bloom, if you do this before flowering your cutting off bloom which if the plant is really overgrown may be OK, just give the customer a heads up to what you are doing. You can also prune burning bushes any time, but I do not shear them, but use hand pruners and reduce size by thining selective branches. Most guys just take a hedge trimmer to everything and are cheaper than I but I refuse the have hedge trimmer will travel mentality.
Forsithia is done after bloom. Burning bush is done late fall. You can prune them back to about the size of a basketball to keep a nice small dense plant. If you want it larger they do take well to shearing (burning bush that is).