Do I need an electrician to move a dental chair?

If moving the chair means moving or adding its electrical supply, yes: that is electrical work in a patient area, and in every state it has to be done by a licensed electrician and tested and certified before the chair is used again. The chair’s circuit is a dedicated, protected circuit sized to the supplier’s spec, so it is not a matter of plugging it in somewhere else.

AS/NZS 3003 also says alterations in a body-protected area should not go ahead unless the area has been tested within the last 12 months. So if the room’s test is overdue, the chair move is the moment it gets done.

If the chair stays on the same supply and nothing electrical changes, it is a mechanical job for the equipment supplier. When in doubt, ask the electrician who tests your rooms: a quick look costs far less than a failed inspection.