If you're using Windows 10, try this solution:
Right Click your taskbar and select “Task manager”
Look for “Intuit update service” in the process tab and end process.
Look for "MSIexec.exe" in the Details tab and end process.
Try to install Turbotax 2016 again.
Run the Windows System File Checker, then try updating again.
Open Start, type: CMD
Right click CMD
Click Run as administrator
At the Command Prompt, type: sfc/scannow
This will check for any integrity violations
Restart your system
Re-register the Windows Installer: Click Start > Run > type MSIEXEC /UNREGISTER > click OK (it'll look like nothing happened). Then do the same thing except this time type MSIEXEC /REGSERVER. Now try updating again.
Start the Windows Installer Service:
Click Start > type Services.msc in the search box > press the Enter key.
Scroll down and double-click Windows Installer.
Select Manual as the Startup type.
Click Start, then OK.
Try updating again.
More about starting and stopping services: