Cleaning a camp stove will depend highly on what kind of camp stove you own. Always consult the owner’s manual (which can usually be looked up online) so you don’t ruin your gear.
Some tips and advice apply to all camp stoves. First and foremost, be gentle with your stove. Use an old toothbrush or sponge, warm water, and soap to clean out the jets.
You can usually use water on burners as well without damaging them. But piezo igniters are a different story. If those get wet, they can fail, and then you will need a lighter or match to start your stove.
When you’re done cleaning your camp stove, air dry it. Allow everything to dry completely before using it again or storing it.