The typical game of D&D is one contain some level of magic. It is easy enough to create a story that requires no magic. It is still relatively easy to populate the world with enemies that aren't magical in nature. However, even if the Dungeon Master keeps tight control over magic items there are still classes like the Wizard and Cleric that are built around magic. The way to handle these classes in a magic-less or low-magic campaign is the topic I hope to cover today (since we can easily remove the magic items and creatures).
The Issues at Hand
I generally don't find it very easy to run magic-less campaigns. The reason for this, however, comes down to handling the classes. Some classes (ranger, I'm looking at you) are very easy to fluff into non magic versions or to simply remove problem spells. Doing that with a class like the Wizards tends not to leave very much (more often it is just easier to remove them entirely).
However, there is another issue at play here. Healing is typically handled by magic classes such as the cleric. Without them, the amount of fighting a group can do is shortened (effectively, the adventuring day is shortened when doing such things as dungeon crawls). This isn't necessarily a bad thing, since it inspires a careful play style and can be very effective when a large portion of the interactions aren't fighting. Regardless, this still remains one of the more difficult issues that needs to be examined when having any group without some form of healing (now, in D&D 5th edition, we have hit die that allow some healing even without a healer).
Removing Magic Classes
We can cut out any class that uses magic in some way. Done. The issue is we are basically left with two (some other rule systems may have more) classes in D&D 5th edition (classless systems or systems with a lot of non-magic classes don't have this problem), and even in those classes not all paths may be valid (the fighter with magic, for example). This can make it difficult to make unique character concepts from the perspective of mechanics unless there is some time spend on other customization options (in D&D, this would be the feats). It can also shift the focus from what players do individually to how their own special builds of the same class function as a group and rotate roles as needed (alright, our lead guy is injured, he goes to the middle and the next in line takes his place).
This method works just fine if the enemies the players will face are not magic. However, without someone who can lift effects such as petrification, fighting a Medusa becomes much more deadly (unless these effects wear off in some way). We also run into the healing issue from above. This, however, can be fixed easily enough by just creating a mundane healing system if there isn't one already (as always, this will need to vary based on the campaign and rule system). This is particularly important in D&D 5th edition, since even the Healer feat won't really help here (in general, it doesn't scale so it becomes quite useless quickly).
Well, if you still want a class that fills the role of cleric without a magic, it is easy enough to just reword the cleric class to sound non-magic, right? Well, for classes like ranger (minus the healing) I find it generally easy to apply this approach except for the occasional spell. However, finding an internally consistent method to reword the Wizard or Cleric is difficult. I've played in games like this before but from the mechanical perspective, it never seemed quite right to me.
Here we are. It isn't exactly rocket science, but at the same time running a campaign in a setting with low-magic or no magic has its challenges. If you were thinking of running this kind of campaign, I hope this helps and if you have anything you wish to share that I missed, please do comment.