It is a farce. A farce is typically the term resevered for comedic plays or films. Farces always have improbable situations, requiring a hefty suspension of disbelief. In addition, farces contain some sort of verbal humor and unexpected twists and turns, as is the case in Goldsmith's play, with all of its confusion and miscommunications.
Moreover, farces typically have some serious intent masked behind their comic fronts. Goldsmith's play, by poking fun at social conventions, like marriage and class issues, made his 18th century audiences consider the problems society faced. Lines like "Ask me no questions and I'll tell you no fibs," also highlight the way society tries to hide from its own actions. Exposing injustice and deceit through comedy is typically more palatable than being preached to if you want people or society to change: think "The Daily Show."