Linden borers make tunnels at the base of trees; very young and very old trees could be damaged. Linden Looper, Basswood Leafminer, Japanese Beetles, cankerworms, Whitemarked Tussock Moth, and the Gypsy Moth can cause defoliation. Mice and voles feed on the stems and could potentially girdle the tree, especially when young. Use a cage to prevent this.
A word on pesticides: Some people may be tempted to use a pesticide as many insects feed on lindens. When the Japanese Beetles descended upon my newly planted Basswood in July, I was indignant. I immediately wanted to go on the defense. But then I remembered that the effect these insects will have on the tree will be aesthetic and are very rarely fatal. This tree is a boon to the wildlife around it.
It is so heavily used by wildlife that the use of a systemic pesticide would be harmful, nay, tragic. In fact, many systemic pesticides state clearly on their labels that they should not be used on lindens (Basswood belongs in this group). Have patience! The insects will eventually move on in their life cycle and in spring the leaves will be good as new. Take comfort in the fact that the tree you have in your yard has contributed to the ecosystem in your neighborhood.
Many diseases can befall the foliage. Anthracnose, black mold, and leaf spot can all affect the leaves, but these issues are aesthetic.
Exposed wood from pruning or damage can lead to canker and yellow cap fungi. Pruning during dormancy can help to prevent this.