It's been a while and I generally try to forget a lot about X-Men 3: The Last Stand, so my memory may be a little bit fuzzy.
Yes, in theory, if he was "cured" or if his powers were nullified for long enough, he would probably die. At the very least, he would be left essentially ill. He would also probably end up having a harder time moving around due to the extra weight of his skeleton and the loss of his enhanced strength.
Would he be cured, or would he be immune?
According to Marvel Wikia, he has
Foreign Chemical Immunity: Wolverine's natural healing also affords him the virtual immunity to poisons and most drugs, except in massive doses. For example, it is extremely difficult for him to become intoxicated from alcohol.
Immunity To Disease: Wolverine's highly efficient immune system, which is part of his accelerated healing factor, renders him immune against all known Earthly diseases and infections.
This means that while Wolverine is susceptible to Leech's powers themselves, he would be immune to the serum as it is just a foreign chemical. Even if he wasn't completely immune, his healing factor would make it so that they would have to up the dosage exponentially in order to affect him, and probably feed him a continuous supply of it in order to counteract his healing.
Marvel Wikia also mentions that Wolverine has been bitten by both a werewolf and by a vampire and that his healing factor cured him of both. Those get into a grey area of whether or not the "curses" are either mystical or blood-borne, but goes to prove the effectiveness of his powers.
Would it kill him?
Again, citing Marvel Wikia - this time regarding his weaknesses:
Adamantium Poisoning: Adamantium has proven to secrete poison which stunts Wolverine's healing factor and feral rage by a very small degree. Even though he has it laced throughout his entire body it has not made a significant change in his abilities. However, the poisoning will endanger Wolverine's life if his healing factor fails on him, such as when an inhibitor field cancels out his mutant powers.
If they were able to nullify his powers for long enough, then yes, it would eventually kill him. But poisons in the blood don't necessarily kill a person immediately. This is proven during the X-Tincton Agenda storyline where Wolverine and many others lose their powers for an extended period of time. Again, it's been a long time - and I recall him feeling ill and even partially crazed, but he survived for an extended period of time.