If the vehicle is pre 1960 then yes it will have to be tested as the exemption only covers vehicles used unladen, if built between 1960 and 1974 then it would have to be tested anyway but Historic tax class can still be applied. Post 1974 it will have to be tested and taxed private heavy goods (if over 3500kgs)
A living van not exceeding 3500kgs is also exempt from testing and plating