Yes, if the nails are big enough to harm the cambium (growth cells) beneath the bark. The nails can open entries into the tree for infection and disease. You also need to consider the size of the nails and the size of the tree. Driving a copper nail into a tree has physical, chemical, and biological actions playing together. If the tree dies, it will not be easy to know precisely the cause.
Copper as a metallic form (i.e. nail) is not harmful to tree's nutrition. It needs to oxidize, to solubilize and then be absorbed. Absorption of copper at the bark level may not occur, while high levels of copper ions in the soil water solution near the root system may lead to toxicity. Courtesy of Dr. Elson Silva, Soil Scientist, Ph.D. Embrapa Area of Science: Agricultural Sciences.