the short answer is no.

in fact as other forumers pointed out, you don't need to unlock your bootloader to root because they are not even on the same level. in a nutshell,
root - gives you full control of your OS
unlock bootloader - opens up the limit of bootloader, which when locked, is only able to load stock ROM
the fact that you asked this question shows you don't really understand what is root and unlock bootloader yet. let me explain more in detail if you don't mind, otherwise you can skip it. 
what is root vs unlocking bootloader?
root
root allows you to add superuser permission which gives you full control of your Operating System (more commonly referred to as ROM in Android community. That's it.
Once you have full control of your OS, you can also take full control of your hardware, such as overclocking, disabling the GPS, etc, but of course you need apps to do so. This is the same with the Folder Mount app.

unlock bootloader
bootloader is a program that loads the ROM (kernel and firmware). It is locked because manufacturers don't want you to simply install custom ROM. When you unlock it, then you can flash custom ROM, but it will be at your own risk because the warranty will be void. You do not need to unlock bootloader if you just want to install app requires more control of your ROM, such as Folder Mount app.

In Sony's case, when you unlock the bootloader, they simply program it to auto-delete the DRM files thus render some apps useless, in the Z Ultra and Z1's case, the camera app too. luckily some XDA developer members have found some way to back it up unlike 2 years ago, there's no way to back it up at all.