When you reset your Hellfire Club campaign or an ongoing Hellfire Club event, you start the campaign or event from the beginning. Any artifacts you had earned previously disappear and combats reset. However, rewards claimed from chests, cleared fog, hidden areas and completed quests do not reset.
You can reset a Hellfire Club campaign as often as you want, and replay it as often as you want. Hellfire Club Events can be reset as often as you want during the event duration.