Most problems with Dogwood trees are related to moisture - either too much, or too little. When overwatered or allowed sit in water, root rot will weaken and kill the tree. Dogwoods do not like to be waterlogged. Plant in a mound if the area is wet.
Fungal diseases are a common problem but can be lessened with winter applications of dormant spray and by watering the soil, not the foliage. Anthracnose can be fatal, but is not as common as it is made out to be. Leaf and bark scorch from excessively warm locations can disfigure the tree.
To prevent most problems, keep them away from standing water, avoid wetting the leaves and plant them in a location that is temperate. That is, not too hot or not too cold.