No, you do not need to include a date on your cover letter. Today, cover letters are primarily submitted online where the date is automatically recorded by either a third-party job portal, email timestamp, or a company’s applicant tracking system.
In the past, when cover letters and resumes were sent by snail mail, candidates would include the date to make it clear when it went in the post.
If you still want to put the date on your cover letter, make sure you format it properly. The correct way to format the date on your cover letter is [Month] [Day], [Year]. For instance, July 29, 2021.