Sheared sills are observed at Piton des Neiges (the eroded basaltic volcano of La Réunion) and are a suspected cause of the coeruptive flank displacement of Piton de la Fournaise in 2007. We performed a 2-D numerical study to quantify the perturbation induced by sill injection within a volcanic edifice. Magma is considered as an inviscid and pressurized fluid injected in an initially stable edifice under gravity-controlled extension. Two cases of injection are tested, in an elastic homogeneous edifice or along a detachment controlled by a friction law. We show that sill injection induces tangential displacements. This effect is strongly increased when sills are emplaced along a detachment, producing extension behind the injection and potentially ending in a large-scale flank collapse. Sill injections can thus explain the cointrusive shear deformation observed at Piton des Neiges and the tangential displacements measured at Piton de la Fournaise in 2007.