So the starter wire is hot all the time? I would suspect a faulty ignition switch. Unless its hot after the relay, then I would suspect a sticky relay, they do stick sometimes.

Then of course there is the possibility that no one likes, that it is shorted with another wire making it hot.

You will have a small arc when you connect it usually. The thing is how small is the arc, is an interior light or something on? Maybe an underhood lamp?