You say it's not your lights & it's not an open door. But the 1st sentence of your post you say "when the key is out of the ignition and I open a door...". Do you mean that the "chime" is not the typical one you'd hear if you left the key in? Will the noise occur if the door is NOT opened (& you simply remove the key)?

"The dash lights up": I take it you're refering to the service indicator lights (check engine, SRS, battery, etc), and not the dashboard (instrument) lights. If this is the case, my 1st guess is you've got some build up of crud in the ignition key switch. I'd shoot into the key slot (this slot is not jammed open, is it?) some electrical parts cleaner. See if that helps. If this was the situation, it would be doing the chiming that you're describing, the same chime as if the door was open, or the seat belt was unfastened.
good luck,
-W-