It's possible that the dye interacts with the protein in a non-specific way, such as by hydrophobic interaction.

If you are using a plate reader to make your measurements, there could also be a measurement artifact. The added protein can cause the meniscus to be deeper, which can affect the sensitivity of detection of the fluorescence. This would not occur with a cuvette-based measurement, however, as long as the meniscus is not in the light path.