I have spent hours researching this and unable to come up with a solution, if anyone knows of one please let me know. I am looking for a way to mount a remote folder on a server (using either WebDAV, FTP, etc) on my Android device as a local folder so that all apps can have access to it and save and read from it as if it were just any other folder on the phone or SD card. Note that I am not looking for a way to "sync" a folder on my phone with a folder on my server, everything in this folder should be stored on the server only and read as if it were local.
On a rooted phone you have several options -- via 3rd party app, directly through a unix
mount command, see How to mount SMB/CIFS Network Shares on android device?
If your phone is not rooted, I do not know how to do that, maybe there can be some app on a play store that is able to mount a network share (NFS, SMB, FTP, ...) via normal privileges (it is definetly possible under unix).
If you want to share just music and/or video you can use DLNA (your phone does not have to be rooted for this to work) -- install a DLNA server on your PC/router and than something like XBMC for Android or BubbleUPnp on your android device.