What is the best time of year to remove bee nests?
Honey bees are different than most other kinds of bees (yellow jackets, wasps, hornets, and bumble bees): Honey bees survive in the winter by staying in their nest, consuming their honey, and huddling together to keep warm. Yellow jackets, on the other hand, have a life cycle basically of:


In the spring, the queen finds a location and starts foraging and building a nest
The first workers hatch, and continue building the nest and foraging, the queen now stays in the nest and just lays eggs. The nest gets much bigger later in the summer, as there are now many more bees.
In the late summer/early fall, new queens are hatched. After mating, they leave to hibernate for the winter (and then repeat the cycle in the spring).
The old queen and all the old workers die in the winter.


So first step is to identify honey bee or not. 

If it's not a honey bee, then you have two choices. If it's not in a spot where you're likely to be stung and they're not really bothering you, you can just wait for the winter for them to die naturally. If you do want to eradicate the nest, then get a can of insecticide and use it late in the evening when most of the bees are in the nest. Spray it into any holes in the nest, and wear protective gear while you do this.

If it's a honey bee, and you do want to kill them, the best time is the evening, and may be easier in the late winter/early spring when there are fewer of them. It may take two or three tries to get them all, especially since the young larvae are protected by wax inside the honeycomb.

Source/more details: http://www.sembabees.org/toplevelpages/bees_in_wall.html



After you remove them, if they were in a wall or somewhere they shouldn't be, then you should figure out how they got in and seal up the holes so they can't get back. Don't seal up holes before they're gone, as otherwise they'll look for another way out, which may be to the inside of your house.