Read the article first linked, it is actually clear enough. The issue is *not* whether or not organ donation is good or bad (in Judaism, it's good, and expected, by definition - it saves lives, and human life is above pretty much the rest of the religion in Judaism).
The issue is about how the organs are recovered, and unlike how the article claims it, it isn't merely an issue with Rabbi Eliashiv being the minority view. The problem is entirely with a conflict between the medicinal and religious definition of death. Judaism generally holds that the heart must stop beating for complete death to occur. Current medical standards hold that brain death is more than enough to recognize death.
So donating organs that don't require you to, you know, continue to function as a living person, those were never in question. The problem is that the donor card gives Israel's medical authorities permission to harvest your still-warm body (has to be as close to death for the organs to be useful). In fact, the problem is further complicated when some doctors bend the definition of brain death, in itself a definition that is not clear across international borders, and which saw particular misuse in the UK. That donor card is basically an encouragement to keep the heart beating and, for that matter, possibly skim the correct definition of death. To balance the need of one dying man against the need of many people needing organs. And that's a fairly problematic and delicate situation.
The matter in Israel would have been resolved by now if medical authorities were willing to reassure potential religious Jewish donors that they won't have their bodies harvested in a matter not in accordance with their beliefs, by allowing the formation of decision boards that would also comprise of Rabbis - said boards would ultimately decide on organ harvesting, which would help mitigate fears religious donors have of having such matters done in a manner against their beliefs.
That's all there is to it. But last time such a board was (almost) formed, medical authorities broke the accords, refusing to let anyone in on deciding such matters as time of death other than themselves.
But the thing is - death isn't just a physical matter. And so long as there are people who don't hold purely physical beliefs, they're unlikely to put their final fate completely and utterly in the hands of those who have a completely different set of core values and beliefs - no matter how well-meaning those people may be.