Nipple chafing or "Runner's Nipple," while somewhat comical, can be mega painful. But you are definitely not alone in experiencing this! It is a common occurrence, especially among men who run with shirts or tops during the hot summer months when they sweat a lot. (Women, although they can still experience it, are not as prone to nipple chafing because of the protection of their sports bra, which acts as a barrier between the skin and rubbing of their shirt or top.)
Let's start with what NOT to do. Do not wear shirts or tops made from course material, or material that does not wick sweat and moisture well. This includes all tops made out of cotton. (Plus, that is just uncomfortable for running in and won't last through time well.) Also, don't continue on running with unprotected and dry nipples. You will just chafe even more.
To first promote healing of the chafed area, make sure you do not run without some sort of protective barrier. A water proof band-aid or soft covering that is taped down will work just fine. This will allow the wound to heal without further exacerbating the chafing. Also, use a thick diaper cream or gel (such as Desitin or Vaseline) to give some lubrication and even more protection.
Once the area has been healed, it is time to focus on preventing chafing in the future. Start by choosing a top (or a sports bra) that is a tighter fit - looser fitting clothing are more likely to rub. Your best option is actually to wear some sort of compression top or gear. Opt for a shirt or sports bra made from material that is soft and fibrous, without seams that run near the nipple. Before you leave for each run, be sure to apply a thick and moisturizing anti-chafe cream or lotion. This will be really viscous and provide a barrier. And be sure to be diligent about drinking water regularly and sticking to a twice a day moisturizing routine. It might be weird to lube up your chest, but it is worth it to be chafe and pain free!