Using a curved capo on a flat fretboard or a flat capo on a curved fretboard may not damage the instrument, but it definitely will not work correctly as previous answers have detailed.
For others who may read this post seeking the answer to the same question who are like me (visual learners), I have provided some pictures below to illustrate the problem that will result from using the wrong capo.
A capo designed for use on a curved fretboard would conform to the fretboard and apply equal pressure against all frets as shown in the image above (exaggerated for effect).
A curved capo applied to a flat fretboard would apply pressure only to the frets closest to the edge of the fretboard and fail to adequately make contact with the center strings resulting in muting of these strings. Diagram is exaggerated to better illustrate the concept.
Conversely (no pun intended), a flat capo on a curved fretboard would fail to properly contact the outer strings causing them to be muted.
The other problem you might encounter is - a typical classical guitar fretboard is significantly wider than an electric guitar or steel string acoustic guitar fretboard. So the length of the capo for acoustic steel string guitar may be too short for a classical guitar.
Just as a classical guitar requires different strings than a steel string acoustic or electric guitar, it also requires a different capo.