This paper concerns the theory and development of inexact rational Krylovsubspace methods for approximating the action of a function of a matrix $f(A)$to a column vector $b$. At each step of the rational Krylov subspace methods, ashifted linear system of equations needs to be solved to enlarge the subspace.For large-scale problems, such a linear system is usually solved approximatelyby an iterative method. The main question is how to relax the accuracy of theselinear solves without negatively affecting the convergence of the approximationof $f(A)b$. Our insight into this issue is obtained by exploring the residualbounds for the rational Krylov subspace approximations of $f(A)b$, based on thedecaying behavior of the entries in the first column of certain matrices of $A$restricted to the rational Krylov subspaces. The decay bounds for these entriesfor both analytic functions and Markov functions can be efficiently andaccurately evaluated by appropriate quadrature rules. A heuristic based on thesebounds is proposed to relax the tolerances of the linear solves arising in eachstep of the rational Krylov subspace methods. As the algorithm progresses towardconvergence, the linear solves can be performed with increasingly lower accuracyand computational cost. Numerical experiments for large nonsymmetric matricesshow the effectiveness of the tolerance relaxation strategy for the inexactlinear solves of rational Krylov subspace methods.