I need help with sweep picking. I cannot mute my notes properly, and when I pull my fingers off a string it rings out, making my sweeps sound like a shitfest. How do you mute ringing strings?
Depends on what kind of sweeps. Some you can use your palm for, just like when palm muting a riff. But really, making your sweeps clean is something that simply comes with a lot of practice and perfection of your own technique.

Good luck
It's in the way you lift off of them. Make sure you're not just pulling off, but allow your finger to have some presence after striking the note (but still stopping the sound) so it'll calm the string down a bit. Some strings will ring regardless though. If it's thAt much of a problem, try getting a girl's hair tie and putting it around the neck of the guitar to stop the strings.

And as always, practice. Practice practice practice.
Generally what most people do is use the right hand to mute unplayed strings above the string (as you're coming down on the sweep) you're currently picking and position their left hand so that the flat of the index finger is muting the strings below the target string being played. For some of the more difficult shapes finger rolling is required where the finger that is fretting adjacent strings must be slightly touching the strings below target string to mute it (same principle with muting on right hand still applies with finger rolling)

Just slow it down A LOT and stay at that speed for at least a week until your muting becomes more natural so you can start focusing on speed.
Tab us out a shape and if I check here again I promise I'll tell you exactly how to sweep it properly; detailed explanation and all.
: )
The string is probably ringing due to the finger release. As mentioned, try and keep the finger touching the string (not fretting-just lightly touching) after you've played that note. Also, mute with the right hand. I don't recommend a hair band, sock, rag etc...it won't clean up your actual playing!