Seat Belt Buzzer Is On When It Shouldn't Be
 

RE:my 1995 E320. Does anyone know what the faulty part would be in this situation?

When the engine is started, the seatbelt buzzer should sound for about 6 seconds or so regardless if you've attached the belt to its receptacle or not. On my car, if the engine is running, and the belt is undone, the buzzer does not stop (it will go on for hours while you drive). The only way to turn it off is to stop the engine and remove the key OR connect the seatbelt. Which part is at fault? Is it the relay that makes that buzzing sound?

Check for a black 2 pin plug at the front left under the seat. It will be coming up through the carpet along with the seat motor cabling. It senses when the belt is in use. The red plug also senses belt use, but it is only used for the SRS computer.

If you cross the black conductors leading to the car (not the seat side) and the buzzing stops, then your buckle is suspect.

Seat belt relay behind speedo?
 

I don't know if your car is like my 90 300E. If it is, when you first open the front doors, are you able to move the seat even without the key in the ignition? Also does the dome light come on when you first enter the car? If you didn't answer yes to both questions, then you need to replace the seat belt relay. It is behind the speedo cluster. See my post from earlier in the month.