You can, and in almost every case you should not. Paint over damp clay seals water in behind a film that cannot breathe, and the piece either peels, bubbles, stays soft in the middle or cracks as the trapped water finally leaves through the weakest point. There is one exception, and it is not really painting.
What goes wrong, mechanically
An acrylic film is close to waterproof once it cures, which is the property that makes it useful on a finished piece and harmful on a wet one. Water still trying to leave the clay reaches that film and stops. It either lifts the paint away from the surface as a blister, works out through an edge and leaves a tidemark, or stays put and keeps the body soft indefinitely. None of these is repairable without stripping the paint off.
The exception: colour mixed into the clay
Colouring the body itself rather than its surface is a different operation and works fine while the clay is wet, because nothing is sealed. Acrylic or powdered pigment wedged into the clay gives an even colour that goes right through, so a chip does not reveal a white interior. It uses more colour than painting does and the shade lightens as the piece dries, so test a small piece first.
If you have already done it
Leave the piece somewhere with good airflow and give it far longer than you would normally allow, turning it so the underside gets air too. Some pieces recover. If the paint has blistered, the honest fix is to peel back the affected film, let the clay dry properly and repaint, because painting over a blister traps the same water again.
Why the temptation exists
Damp clay takes paint beautifully. The colour sinks in, the coverage looks even, and the piece looks finished hours earlier. That short-term result is exactly what makes the mistake common, and it is worth knowing that the good appearance at the time tells you nothing about what the piece will look like in a week.
Questions people ask about air dry clay
Can you paint air dry clay before it dries?
You can, and it usually leads to peeling, blistering or a body that stays soft. Wait until the piece is dry all the way through.
Is there any way to colour clay while it is wet?
Yes, by mixing pigment into the clay itself rather than painting the surface. Nothing is sealed, so drying is unaffected.
How long should I wait?
Until it has stopped feeling cool to the touch and the colour is even, plus an extra day on anything thick.