The general rule of thumb when dealing with any kind of pain is to stop doing what you're doing if it hurts. Beyond that, however, running with hip pain may not be a bad practice if you know what is causing the pain. Simple muscle aches from overuse will more than likely not exacerbate the pain much more, but it can lead to injury if you push the muscles too hard. If the pain is caused by something more serious, or if the pain worsens over time, running with hip pain can cause further damage to an already damaged muscle, ligament, bone, or other essential part of the hip.
Many runners will feel tenderness or tightness after running for long periods of time. This is generally considered normal, and it is often all right to go running with hip pain in this case, as long as the runner does not push him or herself too much. The muscles need time to recover so they can perform their best, but in some instances, a runner may need the muscles to perform at a sub-par level just to make it through the rest of a race. It is important to know your limits; if the pain is too overwhelming, stop running. If the pain is more of a nuisance, and you can identify the source of the pain, running with hip pain can be okay.
Some more serious conditions can cause pain that will worsen if you choose to continue running with hip pain. Small fractures in the bones of the hip can spread and become more severe. Muscle strains can become full ruptures. Ligament sprains can become ruptures, and swelling in joints can lead to a variety of long-term issues. If the pain is debilitating or if it changes your normal running gait, it would be a wise decision to stop running with hip pain. If the pain is slight, or if it is familiar to you — the pain associated with tired muscles, for example — you can continue running at your own discretion. Keep in mind, however, that tired muscles are tighter than fresh muscles, and tight muscles are more prone to injury.
If pain develops after a session of running has ended, assess the severity of the pain before attempting to run again. Try some light stretching to see if the pain worsens or eases, and try walking easily or briskly to see how your hips and legs handle the pressure. While it is best to allow the hip to rest for an adequate amount of time, you may consider running with hip pain if the pain goes away or is generally manageable; keep in mind, however, that you do so at the risk of worsening an injury.