Do Buddhists believe (as do some Christians) in non-material forms of healing? Not necessarily spontaneous, but any type of spiritually assisted healing. 

If so, can you elaborate? 

If not, can you offer a reference?
There are several cases of working cures mentioned in the suttas many more in the Jataka stories which are much later. Two oustanding cases found in the suttas are the one where Angulimala
http://obo.genaud.net/backmatter/appendixes/personalities/angulimala.htm
did an act of truth to relieve a woman of a painful labor, another is the case of the Buddha himself 'bending down his mind' to the suppression of an illness.
http://obo.genaud.net/dhamma-vinaya/bs/dn.16.rhyt.bs_1.2.htm
scroll down to #28

The 'act of truth' is a declaration made to the gods of the truth of some statement in the form of a wish: "If such and such is the truth, let such and such happen." Usually the thing being declared to be the truth must be something hard to believe or embarassing, something not likely to be revealed unless under strong motivation to do so.

Generally the Buddha suggests not working cures with the idea that beings have their kamma to live out. Working a cure does not relieve a person of the experience of his bad kamma, it just postpones it and that may not be a wise idea if it postpones the experience beyond its manifestation in this life there is the possibility that the next life will be so hazardous (as in animal life) that insight and taking counter-measures would be highly unlikely. Consequently curing someone may not be doing them any favor.