I'm not really sure if this is the right category, because I'm not sure what's causing it. Whenever I play higher notes on the fretboard, I start hearing quiet ringing from the other strings. I don't think I'm accidentally rubbing against them, but I haven't completely ruled that out. I'm using a lower-quality amp, so that may be the problem. It happens on both clean and distorted. Any ideas/remedies?
They ring because of vibrations and stuff. Just mute strings you aren't playing.
You may be experiencing some sympathetic resonance.

Time to work on your muting!
As both guys above me mentioned, the other strings are resonating with the frequencies. It's time to really focus on your muting technique with both hands.
Skip the username, call me Billy