Car manufacturers would not include a donut spare in a car if there was a possiblility that it would damage the transmission by using it. Of course, you should follow the manufacturer's recommendation (in the owners manual) about how to use the donut spare. It is only intended to help you get home or to a service center to get the flat tire repaired. These spare tires also have speed limitation written on the sidewall. If you have a flat and use the donut spare as it was intended, you should have no problem.
Many people replace the donut spares with full size spare tires/wheels because they feel more secure, especially aggressive drivers that don't want to sacrifice speed and handling with a spare on the car.