R&J: Why does Shakespeare let us know it's a tragedy? Watch
It also makes the characters more tragic because we know everything they do won't matter, they are caught in a web which they could never have got out of. Their ill-fate is pre-ordained and out of their control, which Romeo alludes to "But he that hath the streerage of my course..." He seems aware that he has been ill-fated. I think being aware the universe is conspiring against you heightens the feeling of tragedy. The allusions to the cosmos or some kind of God already having decided upon the fate of the characters who are described as "star-crossed lovers" again lets us know that their destiny was already written in the stars.
Maybe knowing whether it is a tragedy or comedy gave the audience the opportunity to leave the theatre after they heard the chorus/prelude so they can decide whether they want to stay to watch it or not?
Tl;dr What would be the effect on the audience if Shakespeare didn't tell us it was a tragedy? Would that have created a better experience with more tension for the audience, or would it have detracted from it?