When you mean instrument panel lights, do you mean the milometer digital display, or the back light instrument panel lights that come on when you switch the headlights on. I presume when you remove the key out + door closed, the instrument milometer switches off after several seconds, but the instrument panel back lights remain on permanently. To also state the obvious, although they're switched off on the storks inside, you've checked all the lights outside front and back are actually off.
With the door open and the beeping on, I would remove and replace one at a time fuse numbers 1,2,3,10,11,12,13 and 14 in the fuse box under the steering column. Check too they're the right rating (all blue 15amps). Removing fuse number 10 should definitely cut the instrument back lights, but which fuse(s) stops the warning beeping?
If you haven't got a owners manual have a look at Sarge's pics further down on this post as to what fuse is for what:
Does removing fuse number 10 also stop the beeping. If its another fuse that stops the beeping, do the instrument lights go off too when you remove this fuse
Which ever fuse stops the beeping is the fault the circuits on .
Let us know how you get on.