Code that you write does not need to be licensed under the MIT license. The original code that you received under the MIT license must remain under the MIT license. (Or, maybe not -- but this is certainly true for most licenses, which do not include "sublicense" as a permission. The ambiguous "sublicense" permission complicates this; see Does the MIT license's right to "sublicense" allow me to change the license of someone else's work?.)
So, you must preserve the MIT license and copyright notice as long as you have someone else's MIT-licensed work in your project. Fortunately, the MIT license is very permissive, so it does not stop you from licensing your code (or someone else's differently-licensed code) under other terms alongside the MIT-licensed code, as long as those other terms do not positively conflict with the requirement to display the notice. There are no popular software licenses that would cause such a conflict; such a license would need to have absurdly extreme terms like, "When including this software in a derived work, you must not display notices for any other software included alongside this software." The one category of license I can think of that would cause a conflict is any Creative Commons license with a "No Derivatives" provision, which disallows combining a work with anything else (but that license is explicitly not intended for use with software).
For the sake of clarity, you should clearly indicate in a README which components are MIT-licensed and which are not. The terms of the MIT license don't explicitly require this (unlike, e.g., the Apache license), but if you don't do this, then recipients of the code will not know which parts are covered under which license.