Until such time as our science advances to the point that organs may be grown for the specific purpose of transplantation, organ donation remains one of the most important factors in the medical profession.
Unfortunately, in many cases, demand for organs for transplantation exceeds the supply of suitable organ donors. Under the current system, individuals must specifically decide to donate organs, and many people simply fail to do so, largely as an oversight.
In an opt-out system, people who wish (for whatever reason) not to donate can still have their rights respected, simply by signing the appropriate documentation. But by putting the onus on the donor to opt out instead of to opt in, the supply of registered donors should significantly increase.
Organ donation isn't all that much of a terrible thing as people say. It's only done to people who a) have declined to opt out or have opted in (depending on which program their country supports), b) have organs suitable for donation (patients who have agreed to be organ donors go through tests to see if their organs are healthy enough), and c) are brain-dead (doctors actually put the patients through more tests after brain death to ensure patients actually are brain-dead before transplanting organs). It's not infringing on your constitutional rights or religious beliefs, because if you don't agree with it, then you can opt out.
It is almost forcing people to donate their organs. If people want to donate their organs, they will sign up for it. They should have to opt-out. It is almost taking away peoples rights, they lose the right over their own body, which seems to be taken by the state, which is wrong on so many levels.
Also, choosing to donate your organs can also be a risk. There has been cases where organs have been taken from people who are STILL ALIVE! Here are a few news articles: