It's actually a really interesting question, and to answer it I'll need to give a whirlwind explanation of the Canadian political system, since it is quite intricate and different from the American. It's hard to say really what would happen. The Crown (The Queen of Canada, although for all intents and purposes it really means the Governor-General) is an incredibly powerful position. The royal prerogatives, which are not really written down anywhere, since the Canadian (as well as the British) constitution is partially unwritten and based on convention, include appointing ministers (including the Prime Minister) and dismissing them at will, as well as calling and dissolving Parliament, declaring war, signing bills into law through Royal Assent, etc.

Conventionally, though, these powers are used only through the advice of the Prime Minister, since the Prime Minister has some degree of democratic legitimacy (although interestingly enough, the Prime Minister is not elected directly by the people. The Prime Minister is Prime Minister only because he is the leader of the party that was chosen by the Crown to form government) and the Crown does not, since the Governor-General is appointed by the Monarch, with the advice of the Prime Minister, and the Monarch is obviously a hereditary position. Now, if the Crown were given democratic legitimacy, say through an elected head of state, suddenly the position is empowered and the stability of the system is called into question. Maybe the Crown decides not to sign a bill into law, or dismisses a minister, or dissolves Parliament for no reason, the Crown could say that he (or she) has the mandate of the people to do so, which is obviously bad for the country.

I'm not saying its necessarily bad for the Crown to be empowered like that, just that the system is stable and strong as it is, and it is not broken, so why fix it? It throws a lot of the political system into question and unbalances the delicate balance of our constitution.