Dead limbs cause a variety of problems beyond just the aesthetic. Here are just a few issues:
1) Dead limbs will not heal over well. They typically break off outside of the area (branch collar) where the tree will naturally heal the scar. This leaves a jagged stub sticking out that will start to decay. The decay will then travel into the tree's heartwood and down into the trunk and can kill the tree. This is common in Hackberries since the heartwood is soft and decays quickly. It is not unusual to see large Hackberries that are hollow since the decay has worked its way down from a broken limb.
2) Dead limbs allow carpenter ants to infest the tree. I have seen swarms of them come pouring out of dead stubs that I have cut off. Those suckers sting too. One might argue that if they have a good home in the tree they won't get in the house. There are numerous flaws in this reasoning. If the limbs are cut back to green wood and painted, chances of an infestation go way down.
3) Dead limbs fall. If you are taking advantage of what a nice tree offers - shade, you may have personal items (chairs, table, grill, air conditioner compressor, etc.) under the tree. Ever seen a big limb punch through a glass-top patio table. It aint pretty.
There are other issue resulting from dead limbs. Regardless, take the time to have them removed periodically. This may be required each 3-7 years depending on size and speciest of tree.