During an exit interview, a departing employee accused one of our managers of harassment. Should we investigate even though the accuser is no longer employed here? The manager has been with us a long time, and we’ve never heard any complaints about him before.

Yes, I would recommend investigating the allegations even though the accusing employee has left the organization. If your investigation shows that harassment occurred, I would recommend taking disciplinary action as appropriate.
 
Federal law obligates employers to prevent or stop unlawful harassment. Harassment happens when behavior is unwelcome and based on a protected class such as race, gender, age, religion, national origin, or disability. It becomes unlawful when it is severe or pervasive enough to create a hostile work environment. In this case, since you’ve been made aware of alleged sexual harassment, failing to investigate the allegations could invite risk, especially if additional complaints are made against the same individual.