No, your nipples are generally not removed during a breast lift.
During a breast lift, the nipple and areola are repositioned to a higher, more aesthetically pleasing location on the breast. They remain attached to the underlying breast tissue via a "pedicle," which contains the blood supply (artery and vein) necessary to keep the nipple and areola alive and healthy. The entire breast mound, including the nipple and areola complex, is lifted and reshaped. There's no cutting off of the nipple.
Think of it like moving a house – you're relocating it to a new foundation, but the house (nipple and areola) stays intact and connected to its essential services (blood supply).