I'm considering adding mulch rings around my trees. I really don't want to spend all summer every year picking Weeds out of the mulch though. Would adding the black weed block cloth be problematic? I would assume this would negate some of the fertilization benefits of having the mulch ring in the first place, but thought I'd ask.
There is no way to avoid periodic weed control around a tree.
Any landscaping material that lets air and water through (both necessities for healthy trees) will eventually succumb to weed growing through it. Any material that completely smothers weeds will let neither air nor water through and kill the tree.
You might reconsider your goal to be one of minimizing the effort.
A heavy mulch of wood chips is a good solution. Some people underlay the wood chips with carboard or burlap bags for some additional protection. Burlap lets air and water through quite well but add only a bit of weed protection. Cardboard is pretty effective at stopping weeds and letting air through but not so good at letting water through. With cardboard you may need to water right, soaking right around the trunk, at times you otherwise wouldn't.
Weeds will eventually grow through and upon the wood chips, and, the wood chips will break down over the years. I have found that using wood chips over burlap bags it is very easy to pull the few weeds that make it the first two years, gets a bit harder by year three, and after that needs a new dressing of wood chips.
Be careful not to apply more than 2"-3" of wood chips as this can smother the roots' Also make sure you do not mound the wood chips up around the trunk as this can cover the crown and also smother the tree.
If the tree is surrounded by mowed and maintained lawn it might be time effective to just mow it. I do this on my front lawn with and nice Evans cherry.
Lastly, you could take a permaculture approach and try to establish a local ground cover that is compatible with the tree that will outcompete the weeds and need little maintenance, a subject for another time.
I would go ahead and add mulch to your tree rings. Keep the mulch away from the trunk of your tree. Install mulch no more than 2" in depth. If weeds come up they will be easy to pull. Please do not add plastic that is advertised as 'weed cloth'. The mulch should do a fine job with a bit of maintenance now and then. Only use the finest ground bark you are able to procure. How large is your tree? How large is that tree's ring?