On land here, we have very few poisonous creatures. We do have scorpions, though I've never seen one (and you never hear of anyone getting stung).
We do have bees (no africanized), wasps etc... and we do have fire ants starting to appear. There are a few poisonous spiders (very few actually) such as the brown recluse - again, almost never any news of anyone getting bit. We have big nasty centipeeds which can give a really painful bite - and those you do hear about.
We do have Cane Toads and Giant Bufo Toads - which are not a worry per-say but they are bufo meaning that they excrete a toxin. For people, you would basically have to lick the toad to get the toxin, and it's rarely fatal (actually, 'toad licking' is a means for humans to get high... bufo toxin can induce a pleasant feeling, though I don't recommend trying it, and it is fatal in enough concentration). But dogs can die or get very sick if they lick or bite (if they don't die from it, it usually breaks them of the habit). The Cane Toad can be HUGE - up to the size of a football or basketball (though not as round).
I understand that the Green Poison Dart Frog has been seen on some islands, but I've never seen one here on the Big Island.
Other than an endimic snake called the Blind Snake, which is the size of a worm, blind, and does not bite at all, there are no land snakes in Hawaii (there are water snakes however).
So as far as land goes, even with what I outlined above, the answer is "very little" as most people would rarely see any of the above with the exceptions of bees/wasps and the centipeed (unless your into licking toads).
In the water however, it is a totally different story. The water is chock full of poisonous creatures. There are water snakes that are VERY poisonous, puffer fish, eels, actully - so much that there are a number of books that just cover what can hurt you in the water.
Probably the #1 reported sting would be jellyfish, as that happens every month during certain portions of the lunar cycle (in that, jellyfish will come towards shore, increasing the likelyhood that you would encounter one). The jellyfish can be nearly transparent and dangle long thing nearly invisible lines with barbs that sting. The lines they dangle can go for many many feet (in big jellyfish, they may be trailing lines 8 or 10 feet long). Since they are nearly invisible and may be far from the actual jellyfish, you can easily swim into them and get stung.
You can also get stung by stepping on the lines that have beached, even though the jellyfish is dead. The sting is similar to a bee sting and like a bee sting it keeps pumping toxins even after the initial sting. Unlike a bee, however, the lines have thousands of barbs, so you can get very many stings at once.
Probably #2 would be the variety of anemone (spined slow moving creatures, like sea urchin, etc - the spines can be over a foot long and are like little daggers. while snorkling or scubaing the current can often push you accidently into one - or if your just standing in the water you could acciently step on a small urchin - ouch). On the up side however, sea urching eggs are delicious and are eaten in many cultures including Middle Eastern and Japanese (Japanese call it Uni - mmmmmm, one of my favorites). You can find it in just about any sushi resturant in the US.
There are also a number of fish that are poisonous if eaten (or if eaten without preparing them properly).
There is a GREAT book named "All Stings Considered - First Aid and Medical Treatment of Hawai'i's Marine Injuries" (actually 2 volumns). After reading it I didn't go near the ocean for months.
You mention "insect and animals" - for which I included ocean critters... but you forgot Poisonous Plants... and Hawai'i has TONS. Since we have a issolated ecosystem many things have evolved here that are no other place in the world.
Even common plants - like the mango, which we eat all the time, have a sap which is poisonous (will cause a rash if you get it on you, like a nettle rash)
Castor Bean trees are everywhere, and are VERY poisonous (this is where Castor Oil comes from). Specifically, the seed of the Castor Bean are highly toxic (this is where the toxin Ricin comes from). This is so poisonous that just one chewed-up seed can kill an adult.
Just as with the marine critters, there are a number of excellent books about poisonous plants in Hawai'i.