We discuss symmetries and reductions of the two-dimensional Burgers equation with variable coefficient. We classify one-dimensional and two-dimensional subalgebras of the Burgers symmetry algebra which is infinite-dimensional into conjugacy classes under the adjoint action of the symmetry group. Invariance under one-dimensional subalgebras provides reductions to lower-dimensional partial differential equations. Further reductions of these equations to second order ordinary differential equations are obtained through invariance under two-dimensional subalgebras. The reduced ODEs are then analysed and shown that they belong to the polynomial class of second-order equations which can be linearized only for particular values of parameters figuring in the coefficient.