Not usually. The only way it could cause damage is if a very unlikely electrical problem damaged the SATA port; I have never heard of this happening.
By the way, there are lots of threads here about recovering the data on a failed drive. If your drive still spins, you might want to take a look at any of the dozens of utilities to try to read data from a dying disk.
Edit: It just occurred to me: While a broken hard drive won't damage the mobo, it is possible that both were fried in the same event, like a power surge.