The Monarch of Canada is a ceremonial position. The Monarch's representative (the Governor General) is appointed by the Prime Minister and has no real power (see: the King-Bing Affair for legal precedent), and therefore could technically be done by anyone from anywhere (even beyond the grave). Politically speaking, absolutely nothing would have to change. The Monarch's effective power in our political system would go from basically zero to literally zero, thus eliminating an avenue for potential abuse of power that we risk by keeping a living Monarch as head-of-state. We could achieve this without having to re-open difficult constitutional questions. Traditionalist Canadian institutions with "Royal" in their names (Mounted Police, Army, Airforce, etc) would not have to change their names or branding. Heck, we wouldn't even have to change the designs on our money. Literally nothing would change except closing a loop-hole (albeit a very low-risk one) for potential power abuse in our political system.
The only down side is that smug know-it-alls can say "actually Canada is not a democracy, it's a Constitutional Necrocracy"
Since the Governor General must have real power, it follows that the Monarch must have real power regarding the appointment of the Governor General - rejecting the Prime Minister's request to dismiss or appoint a Governor General when this is clearly an attempt to fill the position with someone who will allow the Prime Minister to act non-democratically.
A dead person will not be able to fulfill this role.