Why do my ears start hurting after running for long periods of time?

Neck spasm? mouth. Could be tight neck muscles, running technique. Could be mouth breathing drying out passages. Do you mean ears or ear lobes? It is a mild winter, but it is winter-wear a hat or ear band. Both ears hurt unlikely to be referred cardiac pain or blood vessel problem. See md to find out.
Probably your jaw. Jaw pain is very frequently misinterpreted as ear pain. Far more likely than ear pain when you run is that you are clenching or even grinding your jaws while exercising. Pay attention to your jaw the next time you run.