iTunes starts playing music without explanation
First step would be to turn off the IR receiver and BlueTooth.

The infra-red receiver can be disabled in System Preferences > Security &amp; Privacy > General > Advanced. BlueTooth in System Preferences > BlueTooth.

You might also try cleaning out the headphone port (gently) with a toothpick and some rubbing alcohol and plugging in a set of headphones.

All of this is to make sure that some extraneous hardware input isn't causing the problem.

If that doesn't fix it then run Activity Monitor and kill the process iTunes Helper and see what happens. If it's something in the System then iTunes Helper will rerun when the problem occurs. If it's a problem with iTunes Helper then killing it should stop the problem. Of course iTunes Helper will rerun when you reboot or run iTunes manually but you will at least get more information.