TL;DR: the organometallic compound created by bonding adamantium to bone at the molecular level breaks down cell walls and prevents the body from making blood & plasma.
It looks like this question already has a decent answer, but it got me thinking - why IS adamantium so poisonous? The comics point out that Wolverine's healing factor isn't just fighting off a slow, gradual degradation or something. It's constantly fighting an infection to the point that the healing factor's overall effectiveness is actually reduced from the effort.
To begin with, I looked up the first "poisonous" metal most people think of - Mercury:
But the real dangerous property of mercury is its density. 13.5 times denser than water, its liquid and gas forms are highly poisonous. When inhaled or ingested, Mercury can accumulate in the body, slowly degrading the membranes of important organs like the brain, nervous system, kidneys or liver. It can cause varying effects from eye irritation and vomiting to DNA and chromosomal damage.
Obviously, this would also apply to adamantium, which would be far denser than even Mercury. But, that's only in its liquid state. Why would the solid form of it cause issues after hardening? Well, let's look at lead - it's considered toxic in solid form:
In the human body, lead inhibits porphobilinogen synthase and ferrochelatase, preventing both porphobilinogen formation and the incorporation of iron into protoporphyrin IX, the final step in heme synthesis. This causes ineffective heme synthesis and subsequent microcytic anemia. At lower levels, it acts as a calcium analog, interfering with ion channels during nerve conduction. This is one of the mechanisms by which it interferes with cognition.
Got that? Basically, it interrupts the process by which the body creates blood & plasma. Still, even with lead it's a rather slow process. A little further research brought to light an interesting fact, however - some of the most toxic metallic substances are formed by a combination of metallic AND organic components together:
In some cases, organometallic forms, such as methylmercury and tetraethyl lead, can be extremely toxic.
And then a few lines down we have:
As a result, [organometallic] metal toxicity has a very high mortality rate even when victims seek timely medical treatment.
So, essentially, the metal itself isn't that toxic in its solid form. In Wolverine's case, however, he has a heavy metal bonded to his bones on a cellular level, which is creating a NEW mineral altogether that is an organometallic compound comprised of adamantium and calcium. That, plus its constant presence without any medical attention, is what makes it so highly toxic to his system.