The classical theory of lubrication is dedicated to the flow of a viscous fluid in a narrow gap between two solid boundaries, with the specific goal of computing the resulting lubrication forces. These forces can be modified by elasticity: the solid boundary can deform elastically and/or the fluid can exhibit viscoelastic stress. Here we focus on the influence of elasticity inside the fluid, using the second-order fluid model, which for steady flows can be viewed as an expansion for small elasticity. We demonstrate how viscoelasticity changes the lubrication forces in various two-dimensional flow geometries, for which analytical solutions are presented. These examples illustrate the physical mechanisms of viscoelastic lubrication at small Deborah numbers. We also discuss the challenges of using the second-order fluid in unsteady flows and large Deborah numbers, and briefly comment on how the modelling framework can get extended to viscoelastic thin-film flows with free boundaries.