If we do full structural relaxation then the structure with defect might change its symmetry. So should we do only ionic relaxation of the defect structure and obtain energy from that?
It depends on your problem. If the defect is supposed to disturb the structural properties of your host material, you should avoid any symmetry constraint in your calculations. On the other hand, applying certain symmetry constraints can give you valuable insights about the competing forces leading to the vacancy formation.
Also you should be careful to consider all possible configurations for whereabouts of your defects/vacancies as well as for you supercell to make sure what you calculate for the formation energy is not for a meta-stable phase.
When a defect introduced to a veteran system it is somewhat comman to get a different symmetry. You have to do full structural relaxation since ionic relaxation don't impact the structure much.
Moreover it is highly depends upon your material...