This article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods is naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both $L^2$ and semi-$H^1$ norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location.