Does an inverter need to be earthed?

The earthing terminal supplied on some inverters is not for electrical safety purposes.

It is intended to be connected to a vehicle chassis to reduce electrical interference to AM radios in weak signal areas, if there is any.

The internal wiring of the inverter is not connected to the metal case.

The inverter does not have any RCD protection, because that is unnecessary with a floating output as used by inverters. There is no connection between the 'Earth' and 'Neutral' pins, and a multimeter will not show 220-240v from the socket when probed - it will be outputting 110-120v per pin.

So, if you do fit an RCD, pressing the 'test' button will not trigger it. As because with the floating output, the dangerous condition that the RCD is designed to prevent, cannot occur.