In the initial chapters of The Count of Monte Cristo, Edmund Dantes explains to Monsieur Morrel that the captain of the merchant ship, Pharaon, has died and in his stead Dantes has delivered a letter to the island of Elba. It is this letter that lands Edmund Dantes in prison. For when he arrested as having conspired to bring Napoleon the exiled emperor who ruled shortly after the French Revolution from Elba, the Deputy Procureur of the King asks him what has transpired. Dantes replies that he has merely followed his dying captain's orders. Upon further questioning, Dantes tells Gerard de Villefort that the Marechal gave him a letter to deliver in Paris, addressed to Monsieur Noirtier. When de Villefort hears this name, he rescinds his orders to dismiss Dantes and says that he must retain Dantes. Then he burns the letter to M. Noirtier in the fireplace.
After this meeting, de Villefort who is a Royalist, serving under the restored king, Louis XVIII, cannot permit Edmund Dantes to mention anything about M. Noirtier to anyone. For, he has great political aspirations. Therefore, he has Dantes charged as a Bonapartist and thrown into prison where poor Dantes spends fourteen years during which time he meets the Abbe Faria, also a political prisoner. Ironically, the abbe was imprisoned for being against Napoleon.
So, while the French Revolution does not have a direct effect upon the narrative, some of the resulting occurrences in the years subsequent to it such as the rise of Napoleon to Emperor and his exile with the One Hundred Days of rule by the Bourbon King definitely affect what happens to certain characters of Dumas's book.