What kind of cleaner to use on a carved Asian chest?

Try linseed or tung oil.

If finish comes off that easy it's a bad finish. May need to be refinished.

I agree with Kelli, if it comes off when you clean it, not matter what you use it may continue to come off, perhaps they didn't use any kind of protective top coat on it. You may need to refinish it.

Is it laquered?? If your lacquered piece is an antique, especially of Asian origin, stick to dry or damp dusting the item and leave polishing to a professional. Never use any type of commercial spray cleaner or spray polish on the lacquer.Sounds like the polish dissolved the finish.????? Not laquer..... Cleaning Repairing Reconditioning Wood furniture https://digitalcommons.usu.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=1505&context=extension_histall

I agree with the steps provided by Lynn, especially this part:

"Moisten a clean soft cloth with paint thinner (mineral spirits) and rub soiled wood to dissolve polish, wax, oil and greasy grime."

Proved to be very effective.