We consider variational inequalities related to problems with nonlinear boundary conditions. We are focused on deriving a posteriori estimates of the difference between exact solutions of such type variational inequalities and any function lying in the admissible functional class of the problem considered. These estimates are obtained by an advanced version of the variational approach earlier used for problems with uniformly convex functionals (see [28,30]). It is shown that the structure of error majorants reflects properties of the exact solution. The majorants provide guaranteed upper bounds of the error for any conforming approximation and possess necessary continuity properties. In the series of numerical tests performed, it was shown that the estimates are explicitly computable, provide sharp bounds of approximation errors, and give high quality indication of the distribution of local (elementwise) errors.