Setting: There are no gods. There is no church. No faith. So that still cling to the notion that the life beyond this world has value belong to a cult or a Wisdom score of 8. The gods have faded away (people watched it happen) or they never existed. Mankind knows for a "fact" that there are no gods. Healing magic does not exist and prayers are never answered. Clerics are a thing of the past.
Rules: Races are whatever the DM wants. However, the class structures will change. For instance, Clerics and Paladins will either go away or change drastically. Cutting them is easy enough. Paladins do not really have a role in game play, although cutting clerics from the game means deadlier combats OR new rules for healing by wizards.
Changing this classes is another thing altogether.
For starters, Clerics and Paladins with FAITH in something will no for a fact that no power is drawn from their faith. However, they can gain different abilities from their convictions. In place of spells and spell-like abilities, the classes would have powers based from their Charisma. Enthrall, for instance, might be an 3rd level ability. It isn't magically, although the DC of the save would be based on Charisma and still be opposed by a Will save. However, the effect could not be dispelled, because it isn't magical.
Clerics and Paladins would also have ROLEPLAYING abilities that most DnD players might find useless, but that would add to the world.
After all, Voltaire said... if god did not exist, it would be necessary to invent him. Therefore, the majority of people would still yearn for religion, even if it were gone. Therefore, religious figures that travel from village to village can still perform weddings, circumcisions, funeral rites, and so on. And... spreading the word of their gospel would be just as important as anything else.
Suggestion would be a perfect ability to add to their repetoire, but it could only be used to give someone new ideas to think about. It would have no real magical effect.
World Logic: Divine Magic does not exist. Faithlessness is rampant. Theocracy is a thing of the past. Knowledge (dead religions) only exists as a skill in worlds that HAD gods before. Knowledge (cults) would replace Knowledge (religion) in all versions of this game world. Of course, perhaps a single god of death remains (responsible for killing everyone), but he doesn't make house calls.
Missions: Nothing really changes here. The PCs are going to play the same game they've always played, although the tone of the campaign and the options of play are going to be different. Try adding the Chaplain class (I wrote) from AEG's Secrets. If you'd like to add something like this.
DM Advice: I ran a campaign once where there was only the SUN and people worshipped it... it actually gave power. Demi-humans had their own cults, but they did not have "Clerics" per se. Only shamans with more "Hearth Knowledge" than actual power.
Witches and adepts are excellent additions to a world without clerics. People with healing "hedge" magic and potions.
Interesting... what if healing is only in potion form?!?