Is there a procedure that can remove the fluid around my lungs without requiring general anesthesia?

Yes. Thoracentesis, as it is known, involves placing a needle into "pleural space" under local anesthesia and with drawing fluid by a syringe or allowing passive drainage. Certain 'rapidly accumulating" effusions may require hospitalization and placement of a chest tube.
Yes. This procedure ( a thoracentesis) can be done with local anesthesia possibly mixed with a mild sedative. If there is a specific reason for the fluid build up, such as a chylothorax, then a more invasive procedure might have to be done under general anesthesia.
Thoracentesis. This is where a needle is placed in effusion and fluid aspirated out. Chest tube can also be used. Can also use pleurx drain for home palliative use. All can be can be done under local anesthesia. Best treatment is to resolve underlying cause, if possible.