If memory serves, Dumbledore was pretty specific that really grave things would happen if the secret got out. If people found out that Dumbledore was sneaking into the girls' dorm ... it probably wouldn't matter; everybody thinks he's insane anyway, and some scribblings in a notebook are not evidence of anything in particular anyway.

However, if Snape in particular found out that Dumbledore had turned Lily against him on purpose because Dumbledore, based on his bizarre worldview, did not think the prim Gryffindor should be with the greasy Slytherin, then it could undermine Snape's secret allegiance to Dumbledore and his secret desire to protect Harry Potter on behalf of his beloved. This would be relatively disastrous.

This is probably already happening, now that Snape has started to see Lily as a human being.