The right rear tail light on my 1989 735i will not stay on - changed the bulb and checked the connections - any suggestions

Bad wiring to the rear tail light cluster.

If the bulbs are ok check for broken cable or bad earth. both tail and number plate lights are inter connected, so if the right tail is ok you have battery voltage to there,

What lamps were you having a problem? So there is a lamp on your dash that says tail gate is open? There is usually a switch or sensor for that?
Maybe I misunderstood your problem?
If the tail gates lamps at the rear are staying on, no matter what you do, I'd have to check wiring diagrams for that lamp control.

check the light bulbs (filiments can break and touch the other filament in the bulb and backfeed other circuits) if they appear ok remove the non working bulbs and swap positions with the working bulbs and see if anything changes

I have the similar situation, while driving got a message on dash that rear lights were not working so proceeded to change bulbs, no solution as problem still persists.someone suggested I buy and replace the rear light fittings and plugs which I did at a huge cost and still the same problem. Any suggestions

Check for power with a test light(not a dvom) at the tail lamp assembly. Test right at the spot where the bulb socket sits. If you have power then i would check the connection tab on the bulb socket itself for pits or corroded tabs. If you do not have power present at test point then double check the fuse chart and see if it says 'electronic' instead of having a fuse amp rating for the tail lamps. If it says electronic for any of the light circuits then it means their is no external fuse that can be replaced and the problem is most likely internal of the headlight switch. Since the headlight switch is actually a power source for the lights. I would start there at this point. Let me know what you find and if you need anything else just let me know

It's almost certainly a wiring issue. Very likely, the ground (-) connection for the tail light has severed or is not making good contact with the chassis. Look for a black wire running from the tail light housing to a screw or bolt on the chassis. You will find that the wire has either become severed, is loose, or is simply not making a good connection. A little WD-40 (penetrating oil) on the screw or bolt where the wire connects will assure that corrosion isn't the issue.

Hope that's helpful!!

CHECK PIG TAIL AT RIGHT REAR TAIL LIGHT IF LOOKS BAD OR BURN REPLACE IF OK FIND FLASHER UNDER DASH AND REPLACE.

It is most likely another faulty bulb in the turn signal circuit.
Ensure the indicator bulbs are replaced with the correct (specified) wattage rating.
Check you haven't dislodged or otherwise lost the ground connection to the rear tail light bulb sockets.

The whole lamp comes out and the bulb is change from the back side.