This article explains why the sign-in box continues to reappear each time you visit Yahoo! on your mobile device.

Your login information is retained by a cookie that is saved by your mobile Web browser. If this cookie is cleared, you will be asked to sign in the next time you visit the page. Some mobile Web browsers clear the cookies and other cached information each time the browser is closed. Also, some mobile devices clear cookies and cached information when the device is powered down or the battery is removed. To determine if your device behaves in any of these ways, please contact your carrier, as they will have the most up-to-date information regarding your device and the way it behaves.

Finally, the cookie that is storing the login information may be damaged. If this is the case, you will need to clear the cache and cookies. Please check your phone's operation manual to get the exact steps to perform this function as steps vary between phones and phone browser applications. Your carrier can also assist with clearing your cache and cookies.