The promise ring is usually worn on the ring finger of the left hand if the ring is meant to symbolize chastity or pre-engagement. If the promise ring isn't symbolic of either of those, it can be worn on any finger.
A promise ring is a symbol of a commitment. The commitment doesn't have to be only one of romantic intent. Promise rings are sometimes used to symbolize a promise to avoid substance abuse or something similar. They can also be used as friendship rings to symbolize a platonic friendship. When the promise ring has no romantic intent, it is usually best to avoid misconceptions of romantic intent by wearing the ring on a finger other than the left ring finger.