I don't generally allow my dog to make retrieves on arrowed (arrow still in) birds regardless of point. I have only had two birds that made it more than a very short distance. So most of the the birds are close and I usually call my dogs off the retrieve.
7 Lakes is spot on with his explanation, but there is still a significant danger with other heads like G5 SGH's or Judo points getting shoved back into the dogs mouth or face. Always try and make your own retrieves if the arrow is still in the bird.
The other option if you really need to shoot broadheads (some need to), is keep your dog on a check cord. If you are running pointers, you can let them work normally but once the dog goes on point, a helper could move in and attach the check cord. Depending on the cover, terrain, dogs, availablility of a helper, and birds, sometimes it can work out very well to use the check cord and other times it can be an absolute headache. Every situation is different.
For slower or lightweight bows, blunts can be very poor options. They simply don't have enough energy to deliver enough shock to the birds system. On one hunt my wife was shooting a 35# @ 28" recurve drawn to about 24" and arrows were tipped with dull two blades. She connected on a flushing pheasant in the backend. The tip penentrated just far enough to wag a couple times and then fell out after a second. The birds flew over the horizon with no sign of injury. If the broadhead was sharp she would have likely brought that bird home. Blunts will never be an effective option for her setup. In those cases, you will just have to make sure you have control of your dogs through check cords and/or proven obedience or leave them home.