Amigurumi (crochet or knit stuffed toys) including felt need to be washed differently than those without felt because the felt is made of a different material, and needs to be handwashed. You can tell if your piece has felt on it by checking for any flat shapes sewn onto it. In the image to the right, there is felt sewn onto both the cap and the body of the mushroom. (Pattern by Critter Stitch Designs.) This set of instructions is only useful to you if your piece does NOT include felt.
I use Poly-Fil, which is made of polyester, as well as synthetic yarns. This means that you should keep your piece away from heat to avoid the plastic melting. Generally speaking, synthetic fibers start to become damaged at about 200 degrees Fahrenheit, so a hot day won't damage your piece, but you should avoid sticking it in the dryer. Instead, use a blow dryer or let it air dry.
I usually recommend handwashing your pieces. You don't have to be picky about your soap, just make sure it's decently mild and that you only use a small amount of it. Literal baby shampoo will work. Wash your piece in cold or lukewarm water, and do not rub aggressively at the surface, as this can damage the material. Generally speaking, be super gentle when washing amigurumi. You can usually get away with using just water to dab away at dirt unless your piece is super dirty.
Because your piece doesn't have felt on it, you can technically throw it in the washing machine, but I wouldn't recommend this as a general rule because it's less easy to tell if something goes wrong. Again, don't put your piece in the dryer afterwards.
If some yarn comes loose, you can use a tapestry needle to weave it back in again.
Unless otherwise requested, amigurumi I make uses plastic safety eyes. Keep these away from small children and pets, because they can snap off easily and become a choking hazard.