No, cover letters do not need an address. Traditionally, cover letters were sent in the mail or submitted in person, so they included both the applicant’s and the recipient’s addresses. These days cover letters are almost exclusively submitted electronically, meaning physical addresses are no longer needed.
Additionally, if you’re applying for a job in a different city or country, including an address on your cover letter (or resume) could put you at a disadvantage when compared to local candidates.