A tarp isn't specifically required but it does have its advantages. Laying out a tarp under your inflatable pool will help to increase the durability of the pool. Twigs and other possibly sharp items hide in the blades of grass and can puncture or wear down the bottom of the pool. Laying out a tarp will add an extra layer of protection between your pool and the ground. Some of the pools on this list are cheap (equally price wise and durability-wise) and it may not be important to you to try to keep intact for next summer. However, with some of the more expensive (and durable) models, you may want to further protect your investment by using a tarp.
Keep in mind that the tarp will be slippery when wet. If you do choose to use one, you have a couple of things to keep in mind. If the tarp is large, it will prevent mud from forming in the yard, keeping it beautiful. However, as mentioned, if it is a large tarp, it will be slippery and the children, or adults, may slip when entering or exiting the pool.