If you've got a dead or dying tree in your yard, don't wait until winter or spring to get it off your property. Have it evaluated now and removed if necessary. This goes beyond trying to remove a tree before winter weather does anything to topple it -- what you're trying to avoid are birds.
Birds nest in trees, and in many areas, native birds are protected during nesting season, which is generally anywhere from late winter to late summer, depending on the region. Most areas are going to be outside nesting season now, so get that tree company over to your property to evaluate and trim or remove the tree. If you wait, you could set yourself up for some nail biting months.
The Migratory Bird Treaty Act
The Migratory Bird Treaty Act of 1918 stipulates that it is illegal to hurt or kill all migratory birds. Many states interpret this to include all species except for one or two non-native species, such as starlings. The Act generally covers nesting birds and their eggs, too.
If you wait until spring to work on the tree, you could find that a bird or two have set up nests. Dead trees are excellent resting places for birds even if you don't like having a dead piece of wood in your yard. Once that nest is there and active, you can't do anything to the tree. In fact, you might not be able to do any sort of construction within a certain radius.
Obviously, homeowners with tall trees are probably landscaping and mowing around nests all the time. But construction and any work done to the tree itself other than normal watering, feeding, and minor trimming, such as of extra blossoms that are nowhere near the nest, can be very disturbing to the birds.
Waiting and Watching
Here's the problem. If you wait, and a nest appears in the tree, then if you find indications that the tree poses a threat to your home (such as a branch that could drop), you might not be able to do anything about the branch if it's too close to the nest.
So, you have two options if you've identified a tree in trouble. Take care of it now, or, if you risk waiting and find a nest next year, keep waiting and keep having the nest inspected to determine when it is no longer active, or when the baby birds have all literally flown the nest.
Why wait? Contact a tree service company to evaluate dead trees now. Consider leaving a snag, or a portion of the dead tree if the trunk is safe (you'll still have to have it inspected periodically), for migrating birds. But other than that, get rid of the portions that could pose a problem before bird nesting season comes along again.