A new class of high-order maximum principle preserving numerical methods is proposed for solving the semilinear Allen-Cahn equation. We start with the method consists of a $k$th-order multistep exponential integrator in time, and a lumped mass finite element method in space with piecewise $r$th-order polynomials and Gauss-Lobatto quadrature. At every time level, the extra values violating the maximum principle are eliminated at the finite element nodal points by a cut-off operation. The remaining values at the nodal points satisfy the maximum principle and are proved to be convergent with an error bound of $O(\tau^k+h^r)$. The accuracy can be made arbitrarily high-order by choosing large $k$ and $r$. Then we extend this temporal exponential integrator to a framework of suitable singe-step methods, such as Gauss method, Radau IIA, Lobbato IIIC. The accuracy with an error bound of $O(\tau^k+h^r)$ is still obtained. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes, which is arbitrarily high-order in time. Extensive numerical results are provided to illustrate the accuracy of the proposed method and the effectiveness in capturing the pattern of phase-field problems.