Cell-shape changes are insured by a thin, dynamic, cortical layer of cytoskeleton underneath the plasma membrane. How this thin cortical structure impacts the mechanical properties of the whole cell is not fully understood. Here, we study the mechanics of liposomes or giant unilamellar vesicles, when a biomimetic actin cortex is grown at the inner layer of the lipid membrane via actin-nucleation-promoting factors. Using a hydrodynamic tube-pulling technique, we show that tube dynamics is clearly affected by the presence of an actin shell anchored to the lipid bilayer. The same force pulls much shorter tubes in the presence of the actin shell compared to bare membranes. However, in both cases, we observe that the dynamics of tube extrusion has two distinct features characteristic of viscoelastic materials: rapid elastic elongation, followed by a slower elongation phase at a constant rate. We interpret the initial elastic regime by an increase of membrane tension due to the loss of lipids into the tube. Tube length is considerably shorter for cortex liposomes at comparable pulling forces, resulting in a higher spring constant. The presence of the actin shell seems to restrict lipid mobility, as is observed in the corral effect in cells. The viscous regime for bare liposomes corresponds to a leakout of the internal liquid at constant membrane tension. The presence of the actin shell leads to a larger friction coefficient. As the tube is pulled from a patchy surface, membrane tension increases locally, leading to a Marangoni flow of lipids. As a conclusion, the presence of an actin shell is revealed by its action that alters membrane mechanics.