My roommate often uses Tiger Balm and sometimes browses through my collection of vintage Star Wars figures. Can the residue on his fingers cause harm to the PVC figures or cause the paint to crack or chip?
According to this Wikipedia entry on Tiger Balm, there is camphor (among other chemicals) in the formulation of this product. This chart of chemical resistance recommendations for PVC indicates that it is not recommended that PVC should be exposed to camphor. Therefore, it is probably best to reduce or eliminate contact of Tiger Balm with the figures.
As for the paint cracking or chipping as a result of exposure to Tiger Balm, that might depend in part on the composition of the paint, and that cannot be addressed unless you can provide that information.