Though Microsoft claims that installing Windows 10 Technical Preview Update Rollup: February 4, 2015 will fix this issue but even installing the update a user may experience the same unresponsive Start Menu. The problem not only makes the start menu unresponsive but it also make the Start Search non functional.
System Restore or running sfc doesn’t help. I have resolved the issue by the following way:
Sign out from your Microsoft Account and reboot the system with a local account (if you have made earlier). You will find the Start menu has started working again. Now you can switch over to Microsoft account.
You may create another user account with administrative privileges.