I have a 2010 Forester.  So even though you remove the key from the ignition the door chime constantly goes off with the door open.  the head lights do not turn off and the doors will not lock with the power locks.  outside the car the remote transmitter will not lock/unlock the car  also the alarm will not arm.  I'm guessing the car still thinks the key is in the ignition but why?

In most ignition locks there is a little switch that's activated by the key as soon as it's pushed into the lock cylinder.

Sometimes those switches can get stuck due to dirt/dust/ crud getting in the lock.

Get yourself a bottle of lock-ease. Most hardware stores have it. Its a special oil with graphite in it. Squirt a bit of that in the lock, and put a drop on the end of the key. Put the key in and out of the lock about a dozen times and see if that fixes it.