Usually if the neck is broken, the bird is paralyzed and can not move. Very sick birds and birds with some infections often have a limp neck. Most infections would not transmit to humans, but some might. Make sure you pay attention to hygiene and wash your hands after handling the bird.
You are likely to cause more stress and harm by trying to stabilize the neck with a brace or bandage. If you want to try feeding the bird, try to control the bird using a towel and stabilize the neck and head with one hand.
I wish you and the dove the best. Let me know if you have follow up questions.