There are many animals that live on or near volcanoes. It depends on where the volcano is located to determine what lives on the volcano. If the volcano is in a warm tropical place there will be animals like monkeys and birds and plants like palm and fruit trees. If it is in a cold environment there will be things like polar bears and evergreens.
Directly after an eruption there will be next to nothing left on or near the volcano. The thing is after an eruption soil becomes very rich making it easy for new plants to grow. It won't take very long for trees to start growing after an eruption. When there is enough plants herbivores will start to live on the volcano again, then after there are enough herbivores for them to eat on the mountain many carnivores like mountain lions, bears and coyotes will start to come.
Now you are probaly wondering why soil becomes so rich after an eruption. It becomes rich because ash will cool and eventully turn into fertilizer. Magma also helps by breaking down and feeding the plants. Here are some some animals that live on or near volcanoes in a mild environment, like here in B.C.
Here are some animals that live near colder vocanoes (like in Alaska and Antarctica)