Can horses run for long periods of time?

Most of them.

Another Answer:

Some breeds are able to run for much longer periods. These are mostly Arabians, the Akal-Teke, and similar breeds.

These particular breeds are commonly used for endurance races. Some breeds like the draft horses (Clydesdale), are not built for running, so they cannot run for long periods. It also depends on the horses training. Horses are athletes and just like human atheletes they need training, exceptional diet, and conditioning to be in top shape to run for long periods.