I have an Acura Integra and I recently replaced one of the burned out tail lights. Then all of a sudden the interior dashboard lights, lighting on my shifter and brake lights do not light up unless I brake.

 

Not up to speed on Acura thus can only guess.

Guessing when you brake the electrical power going to the brake lights is feeding back into the other circuits.

On a Camry this might indicate a short between wires in the harness, typically in the trunk where part of the harness goes to the trunk lid. The wires in this area can break do to flexing by the trunk lid action.

Or the wiring around the lamp that you replace is damaged.

On rare occasions a bulb could have problem, sometimes in a dual filament bulb one filament end breaks and part it fails onto the other causing strange problems.

Suggest you check for bad wiring starting in the area of the bulb you changed. Also replace the changed bulb with the old one or remove it and check what happens when you brake.