Hand painted furniture needs special care given in order not to ruin any of the painted features. In general, a mild soap and water mixture can be used for most hand painted furniture that has been sealed. You should stay away from toxic chemicals that can eat away at the paint.
Two words: Be Careful! To clean, wipe down with a damp towel and quickly follow with the dry version. Always use coasters, dust often, and use a sealing spray if possible to prevent wear and color-changing.
You care for hand painted furniture by making sure you clean it with just water. Make sure the cloth is damp and has a few drops of water on it as well. Thus, that should be sufficient in just cleaning and taking care of the furniture. That is perhaps the best thing.