Mushrooms are neither animal nor plant. You may think the mushroom would be a plant but they do not have the chlorophyll a plant requires to make its own food. A mushroom is the fruit of the fungal organism that produces them. By picking a mushroom you do not harm the fungus itself, because the main body of the organism lies underground and grows quietly and unseen for most of the year until the conditions are right for fruiting which is when mushrooms appear.
Mushrooms are not only fascinating to look at; they are the main recyclers in nature. They break down wood and humus into their original components and provide food for living plants by returning dead trees and forest litter to simple organic materials in a form suitable for plants to use.
Some mushrooms have a symbiotic relationship with other plants. This beneficial association may be expressed by a tree giving up some of its glucose to the fungus while the fungus gives some of its minerals to the tree. This is why some mushrooms are always associated with certain trees. One group of symbiotic mushrooms grows underground and can only be found by digging for them, but to even find them requires specially trained dogs or pigs. An example of these would be truffles, which are very expensive and sought after by chefs the world over.