The major result of Ophelia's madness could rightly be assumed to be Laertes willingness to go along with Claudius' plan to kill Hamlet. They arrange the duel, the poison is placed on the sword and on the pearl, things are all arranged and everyone, in the end, dies. Without Ophelia having completely lost it and her brother seeing this and flying into a rage, it might not have been possible to convince him as he had some respect for Hamlet.
But another effect is the emboldening of Hamlet to try and act. It is clear from the scene around Ophelia's funeral that he grows more capable of voicing his feelings, of making things public and the action proceeds from there.