If the tasks are independent (don't depend on data from the other tasks), and can be performed independently, then yes, by all means execute them asynchronously. If they're Entity Framework, and it's version 6 then it offers Async methods to call, and you don't have to wrap them in Task.Run.
Even if the tasks aren't independent, you may still be able to order them in a way to make them more efficient when executed.
Whether or not to use Task.Run, however is an important distinction. Task.Run will use a ThreadPool thread, which if it's doing something synchronous will block, and therefore reduce the number of ThreadPool theads available to your application. If you have many users, and you are executing many tasks, this could be a problem.
Try to find async api's rather than using Task.Run.