Use Paper as Shim for Circuit Board in 3V device
Paper won't cause a fire*, however there's a (small) chance it could affect the operation of the device due to leakage. 

Almost any kind of plastic would probably be better (be a bit suspicious if it's black color because occasionally it's designed to be conductive). Just eyeballing what's around me and subtracting what is techie-specific (so much stuff), I see plastic covers on notebooks that could be easily cut with scissors. A 'Mead' 5-star notebook cover is about 0.5mm thick (0.02") and looks about perfect. Some thin binders or report covers are also a possibility and should be available cheaply in a dollar store or office supply store. 

If you have any ancient overhead projector foils, that would be another one. 

Rubber bicycle rim tape is another possibility (the kind that goes inside the rim to protect the tube). Put the adhesive on the side away from the PCB. 

* Note that paper will not directly cause a fire, but it will sustain a fire so you would not typically find untreated paper in a real product that demands UL94 V-0 or similar material ratings. Many plastics such as polyethylene are also quite flammable. Often fire retardant additives are used to make plastics acceptable, can they can cause other problems (some are carcinogenic). In any case, that's not of too much worry in a device powered by AA alkaline cells.