After a swarm of bees either clears out of a hive or is removed, you may end up with a piece of the hive still attached to the tree or building. When the insects are gone, the empty hive structure may seem harmless, but in some cases it's best to remove the entire structure. It's also essential that you don't try to remove the structure yourself in case you get a nasty surprise.
Proximity to Home
How close is the hive to your house or to any structures that you or others use? A hive that's off in the woods away from where you and other people usually go might not be such a problem for you. A hive that's on your house, though, should definitely be removed. If nothing else, the sight of the structure can give you a jump when you see it. Removing the hive also avoids the possibility of the hive being rebuilt.
It is possible for a new swarm to take over an old hive. Even if the hive has been broken open or is just a small mass with most of the body missing, bees can build upon the structure and create a new hive, which starts your bee problems all over again. What's worse is that you never know if the new set of bees is particularly aggressive. Saving local bee and imported honeybee populations is important, but you don't have to do that at the expense of your safety. Remove the remainder of the old hive so that you don't have to worry about it becoming active again.
When a swarm vacates a hive, all the bees should go. But you can never be quite sure what's currently in the hive, such as a few leftover bees. This is why you want a pest control company that works with bees to remove the hive. If you do it, and you aggravate a bee or two that might still be in that hive (or new bees investigating the structure), you would then be at risk of stings. Bee control services will have the right protective gear and know the best way to remove a hive without creating a problem.
Of course, if you're comfortable with the hive being where it is, you might not feel the need to remove it. Still, proximity to the home and the potential for a new active hive may make removing the hive a lot more attractive to you.