The Next.js App Router has several common mistakes that developers make when building with it, including using Route Handlers with Server Components unnecessarily, not providing absolute URLs for fetch requests, and not revalidating data after mutations. Additionally, there are issues with using Context providers with App Router, adding "use client" directives unnecessarily, and handling redirects in try/catch blocks. The App Router offers new features like Static Exports, Partial Prerendering, and Server Actions, which require a better understanding of the model to take full advantage of them. To avoid common mistakes, developers should learn about these concepts and how to apply them correctly in their Next.js applications.