Whenever I play a piece, sometimes when I lift my finger off the note I've just played, the open note of that string will slightly be audible. Can this be stopped, or do I have to learn to control it?

I have tried to control it but it's very hard.
I think you release your finger too quickly. Remember that the faster you release the finger (and the more motion you give it) the more it will sound.

Try releasing super slow, and then faster and faster until you still are slow enough to mute the string.

That's how I do it at least.