Can you paint furniture with distemper paint?

You can and many people do, however a homemade soft distemper alone will easily wash off even when dry and is difficult to seal. Commercial distempers can be sealed with a water based polyurethane, but will not provide a strong bond and could still eventually delaminate from the surface.

If, in the future, you want to refinish the surface with another decorative treatment, it's recommended that you remove the distemper layer to allow for a tenacious bond on the new coating.