This is a great question because there are, indeed, several elements of Restoration comedy in Oscar Wilde's The Importance of Being Earnest.
The first Restoration element that can be easy to identify in the play is the combination of different plots. Restoration comedy, which begins in the decade of the 1660s- 1670s, parted away from the French comedies of Moliere, for example, which were simple enough in plot to focus on one character or one situation that was both picaresque and witty. The English audience's hunger for more complex plots, even in the comedic genre, led to the creation of aristocratic comedies that mocked several different aspects of the life of the aristocrats to include marriage, personality, social life, and politics.
The Importance of Being Earnest has those complex plots: One could tell a complete story about the lives of every single character in the play, because they are that deep and complex. Moreover, their depth is what brings, with each of them, new problems and situations that become tangled throughout the plot.
Another Restoration element is the lack of emotion that is seen in some of the characters. This is done so that the unique quality of the character remains strong and untouched throughout the play and not intermixed with emotions that might end up changing the nature of the character.Sure, we know what each of the characters like and what they do to get what they want, but Algernon (for example) is obviously oblivious and careless, Cecily and Gwendolen only fantasize about love, Lady Bracknell and Miss Prism seem a bit frigid to a point, and (with the exception of the end of the play) we do not see a true romantic inkling in Dr. Chausible. If anything, it is Jack who seems to have emotions for Gwendolen, but we also wonder if they are being properly compensated.
One last element that is obvious is the lack of heroes, and the focus on weaknesses and anti-hero qualities. Jack is not the prude and flawless man that one would expect to see in heroic dramas or comedies. He leads a double life, just like Algernon, and he obviously likes to get in trouble as much as Algernon does. He openly expresses his hatred toward Lady Bracknell, and he admits to the social flaw of not having a family name that he can claim as his own. Algernon, on the other hand, is an anti-hero, hands down. He cares nothing about anybody but himself and, as long as he can eat excessively, he is happy.
Therefore, the complexity in plots, the lack of overall emotion, and the focus on flaws and weaknesses are some of the most salient elements of Restoration comedy in Oscar Wilde's The Importance of Being Earnest.