If I were to make transcriptions of copyrighted music and make them available on sites such as MuseScore or post the audio on YouTube and I did not make any kind of advertising revenue or any money selling the transcriptions, would I face any kinds of legal problems?
Yes, that would probably result in a copyright violation. You're making a copy of the work, albeit in a different medium.
The fact that you make no money wouldn't do much to help you. The question is not really whether you're making money, but rather whether you're cutting into the artist's ability to make money. Giving away copies for free is a pretty good way to do that.