When playing pizzicato on the Violin (specifically on the A and E strings), I hear the D string vibrate as well, even when I don't touch it. This causes the notes I actually play to sound a bit weird sometimes and it annoys me. Why does this happen, and how can I avoid this?
These are called sympathetic vibrations, and they happen because the note you're playing is an overtone of one of the open strings. This can be demonstrated very reliably on a properly tuned piano: slowly press one key so that the damper is off but you don't sound the note, then while holding that slam the key an octave lower. The first note should ring quietly.
I suspect that the sympathetic vibrations you're hearing are far too quiet for the audience to hear. If you want to stop it, the only thing you can do is use other fingers (on either hand) to mute the strings. Guitarists do this kind of thing all the time.