That would depend on the design of the instrument. The refractive index of a 12 °P sucrose solution is 0.7% greater at 430 nm than it is at 700 nm so presumably if you had an instrument designed for 700 nm it would read 0.7% higher if you used it with a 430 nm light source. But a sensible design would include a filter in the cover or coating on the prism surface to pass only a relatively narrow bandwidth of light i.e. one centered at the wavelength for which the instrument is designed.
The main difference between natural light, tungsten light and fluorescent light is that the first two have continuous spectra whereas fluorescent lights have line spectra (most of the energy is centered at a few wavelengths). Thus the possibility exists that you would have a refractometer designed for one end of the spectrum being illuminated with a light source which has lots of energy at the other end. This would broaden the perceived transition beween light and dark but I don't see how the error would get bigger than 0.7% (e.g. 12.08 °P for a 12 °P solution).