Oriental bittersweet berries start to form in late summer into fall so the best time to cut vines is before the berries develop. If you cut after the berries develop they'll still drop from the cut vines and re-seed the area. Birds will also eat the vines and spread infestation to your neighbors and around town. If you don't cut at all that guarantees the berries will develop so if the only time you can cut is during berry season, go for it. It will help limit the infestation next year.
From a comfort point of view springtime is a good time to cut, before weather gets too hot and before extensive underbrush develops and makes access to the base of the vines difficult. Cutting before leaves develop on the vines will allow the tree canopy to benefit from spring growth and begin recovering sooner.
Keep in mind that if leaves have not appeared yet you may not be able to distinguish small bittersweet vines from poison ivy, so take appropriate precautions on the assumption that some vines may be poison ivy.
If you are taking a "cut and treat" approach an excellent time to cut is late fall when the vine is pulling nutrients from the leaves to the root system. It will draw any treatment you apply to the cut stem back to the root system. Whether or not to use an herbicide is an individual decision (see "Invasive Plant Control and Disclaimer" tab of this blog).
Although some times are better than others there is no bad time to pull or cut oriental bittersweet vines. Cutting even one large vine about 9" above ground and then cutting the portion hanging from the tree about 5' off the ground will provide a huge amount of relief to the tree canopy as the vine drops its leaves and stops strangling the tree trunk and branches.
Cutting mature vines a little above ground level makes it easier to go back later, find the cut stems and inspect for re-sprouting. Pulling the entire vine out of the ground is an excellent approach but is not always possible. Cutting the portion left hanging from the tree about 5' above ground makes it harder for future vine regrowth to latch onto the old vine and climb back up into the tree.
Pulling the remaining vine down from the tree is a good cosmetic approach but not necessary. Most vines are quite securely attached and pulling the vine can damage tree branches. It can also be dangerous if the vine is attached to broken or rotted tree limbs. If the vines have leaves on them you may not be able to see the condition of the tree limbs under the leaves. Eventually the vine will drop on its own.
Properly dispose of cut or pulled vines, especially if there are berries on the vines. For proper disposal information go to CIPWG's disposal guidelines.
Be sure to do your research before attacking your bittersweet infestation. It will make your investment of time and work much more effective. Always inspect trees and the tree canopy for unsafe conditions before working in any wooded area. Identify any poison ivy (use a product like Tecnu), and inspect for ticks after working in the woods.