The moral of a story is the lesson that a reader can learn from the story or characters and apply to life. All throughout Dumas' The Count of Monte Cristo, Dantes evolves. He goes from a young, naive boy, to an educated and vengeful nobleman, to a more wise and compassionate man. One lesson might be that even though life throws various injustices at a person, he or she can overcome everything with education, a good plan, and persistence. However, there might be a better moral because Dantes realizes that revenge is less satisfying than he hopes. After 14 years in prison, and then about a decade educating himself and executing his revenge, he finds himself alone and unhappy. This does not mean that he has not learned anything along the way, though. In fact, Dantes and Morrel discuss the injustices of life in the end, and it is here that the moral of the story can be found:

"'It takes a long time for eyes that are swollen with weeping to see clearly, and at first, perhaps, he did not comprehend this infinite mercy, but at length he took patience and waited . . .'

'Is it possible for this man ever to be happy again?'

'He hopes so.'" (607).

For Dantes, being patient, waiting, and hoping for happiness might be the best that a person can hope for when facing the injustices of life. He also comes to understand that "infinite mercy" can also play a role in life, not just seeking satisfaction for the wrongs inflicted upon him. (Carrying a grudge for multiple decades can really run a person to the ground.) The moral of the story is repeated in a letter from the Count to Valentine and Morrel as follows:

"Live and be happy, beloved children of my heart, and never forget that, until the day comes when God will deign to reveal the future to man, all human wisdom is contained in these words: Wait and hope!" (617).

Above, the words "live and be happy," and "wait and hope," put forth friendly advice for a life that Dantes wants for Valentine and Morrel. For most of his life he waits and hopes for a happy life, but in the process, he doesn't live happily because he is too busy plotting revenge. Moving forward, though, Dantes hopes to be able to live that happy life with Haydee that has eluded him so far. Fortunately, readers can also take Dantes' final lines and apply it to their own lives as the moral lesson of the novel—"Live and be happy . . . Wait and hope!"

I think the first answer is great, I would add just one thing.  The early picture of Dantes as a trusting, hard-working and loving man is an interesting contrast to the Edmond we see throughout most of the story.

He changes into a rather evil, incredibly manipulative man who trusts absolutely no one, except for perhaps his trusted servant.  Because of this, his life is filled with wealth and power and eventually revenge, but none of the happiness and hope and excitement for the future he felt at the outset.

So the idea of being able to trust, of having plans and hopes for the future besides the horrible death or ruination of others is an important part of the moral of the story as well.

The lesson that Dumas makes quite clear in The Count of Monte Cristo is that pure revenge does not satisfy.  While most readers would agree that Dantes has every right to be angry and to seek justice, Dumas strives to show that despite all the effort that Dantes puts into getting revenge, in the end that is not what satisfies him.  Rather, he gains contentment from restraining himself from harming or using those who have become close to him and by being willing to "walk away" at a certain point.