The area above-treeline is called the Alpine Zone in the White Mountains, and while it looks like a bunch of rocks, there are quite a few plants that grow there. Many of these plants are rare, and have very short growing seasons because they’re covered in snow most of the year. Camping is quite disruptive and harms the plants because people walk, lie, and cook on top of them. Imagine if someone broke the branches off some of the small plants for firewood because they wanted a campfire. They’d never recover in the short growing season they have and die.
Therefore, camping above treeline in the alpine zone is forbidden by the US Forest Service which oversees the White Mountain National Forest except when there are two feet of snow (or more) on the ground. This is enough to protect the plants from human disruption during the winter.