Error propagation and perturbation theory are well-investigated areas of mathematics dealing with the influence of errors and perturbations of input quantities on output quantities. However, these methods are restricted to quantities relying on real numbers under traditional addition and multiplication. We aim to present first steps of an analogous theory on idempotent semirings, so we define distances and norms on idempotent semirings and matrices over them. These concepts are used to derive inequalities characterizing the influence of changes in the input quantities on the output quantities of some often used semiring expressions.