The invention discloses a variable-height bending shear type eccentric support energy dissipation beam and an eccentric support structure, and belongs to the technical field of steel frame support structures. The energy dissipation beam comprises an upper flange, a lower flange, a web plate and a transverse stiffening rib. The web plate and the transverse stiffening rib are perpendicularly arranged and simultaneously connected with the bottom surface of the upper flange and the top surface of the lower flange. The web plate comprises a first area, a second area and a third area which are sequentially connected in the length direction of the energy dissipation beam. The height of the second area is less than that of a web plate of a non-energy-dissipation beam. The upper flange and the lower flange are matched with the web plate in structure. The width of the upper flange is less than that of an upper flange of the non-energy-dissipation beam. The width of the lower flange is less thanthat of a lower flange of the non-energy-dissipation beam. By arranging the second area with the low height on the web plate, after the two ends of the energy dissipation beam are subjected to bendingplastic deformation, the middle of the web plate can continue to be subjected to shear plastic deformation, the energy dissipation effect is improved, and anti-seismic performance is enhanced.