Turns out you don't need a special app on your phone or tablet to control VLC remotely — it comes with a HTTP interface that's easy to set up and can be accessed from any device that can connect to your network.
To enable the HTTP interface, make sure you have the latest version of VLC — at least 2.0.0 or greater. It can be enabled on earlier versions via the command-line, but there's no need for this in 2.0.0+ as it's built right into the Preferences window.
Start VLC and select Tools -> Preferences and check the "All" radio button near the bottom-left corner of the Preferences dialog. Then, in the left-hand tree view, select Interface -> Main Interface and check the box that says "Web" in the "Extra interface modules" section (see the right-hand image).
Leave everything else as it is, save the changes and restart VLC (this is important, otherwise the web interface won't be available). Open a browser and type "127.0.0.1:8080" into the address bar, or "localhost:8080" if the first doesn't work. You should see something similar to the lead image.
To access this interface in a convenient manner on your other devices, it's best to set a static IP for the machine running VLC, otherwise you'll have to change the address each time your router hands the VLC machine a fresh one. Once you've set the IP — for example, to 192.168.1.5 — you can visit the interface by typing "192.168.1.5:8080" into the address bar of any browser — be it on a PC, mobile or tablet.
Documentation:Modules/http intf [VideoLAN Wiki]