How Do I Remove Instrument Cluster 71 Dart?

Assuming speedo cable is disconnected. Referring to the pic you added... I wont say removing the radio is the best plan but.. If you did remove the ash tray and radio you would find the wiring harness captured under those 2 metal straps and the harness connector on the wiper switch in your path to the inst' panel. Work your way around those and the 2 wires on the amp gauge and the inline connector on that circuit board could be removed.
These inline connectors are notorious for causing bent, loose, or broken pins on the circuit boards. The one on your left side board is close to the top so the panel tilting is stressing it. Find a path to them and carefully pull them straight away from the board is definitely the best plan.
When I remove any a-body inst' panel, I remove the one screw holding the fuse box so it and its wiring will move. The I can easily push the little button on the headlight switch body to remove its stem. I remove the knob from wiper switch. I remove both switches bezel nuts so switches can fall inside as far as the wiring and dash will allow.
I don't know that all this would help you since your switches aren't mounted in the inst' panel like most other examples. I do know you will generate more movement in this hard old wiring, plus you will have a path by the fuse box to the speedometer cable to disconnect and reconnect it.
When the factory service manual was written, "pull the inst' panel outward and tilt it", They weren't fighting a 50 year old wiring harness.