The most potential for damage to young or thin bark trees occurs in the spring and early summer. During this period the bark is easily peeled and damage to the cambium layer is a problem. The tree's cambium is composed of a layer of cells that transport nutrients and water to the tree. This layer is pretty fragile until later in the summer. While a hammock anchor, or other line around a tree, may not completely girdle the tree causing it to die; the wound does disrupt water and nutrient flow to part of the tree, and is an entry point for pathogens that cause disease and decay. This is mainly an issue with younger/smaller trees with thin bark, but can be a problem with some larger trees. Where I've really seen this to be a problem is in Wilderness Areas where horse people tether their horses to trees.
One mitigation is to use a wider webbing instead of a rope or line, which will distribute the load around the tree's circumference.