Understanding and controlling how carbon nanotubes interact with phospholipid membranes is necessary for preventing adverse effects of these relatively new, but still exciting, materials. Futuristic applications envision incorporating carbon nanotubes in liposomes for personalized medicine, controlled delivery, and imaging. Because of their ability to penetrate phospholipid bilayers, nanotubes could serve as nanoscale syringes to deliver molecular cargo and develop gene therapy. Several experimental reports available on the subject demonstrate the need for a better understanding, at the molecular level, of whether carbon nanotubes penetrate, reside, and perturb phospholipid bilayers. Using all-atom molecular dynamics simulations, we quantify how short carbon nanotubes (~6 nm in length) embedded within a DOPC phospholipid membrane perturb the structure, organization, and dynamics of the lipid molecules within the membrane. It is found that the structural perturbation is very short-ranged, although it becomes pronounced when bundles of carbon nanotubes are formed within the membrane. The presence of the nanotubes is found to reduce the mobility of lipid molecules within the membrane and to perturb the structure of interfacial water. Our observations suggest that the local perturbations in the lipid structure caused by the nanotubes could lead to enhanced penetration of molecular compounds across the membrane.