This is something I’ve been wanting for a long time, and Volumio is the first platform that I think could handle it.
Create an alternate web interface with a single button. This button basically acts like a down-vote. The trick is that the server keeps track of how many unique browsers are connected to it, and considers the down-votes as a percentage of those connected. When the downvotes hit a threshold, it skips to the next song.
The downside to this is that it may be like an elevator button for the impatient. Some people will wind up pressing it when they’re bored. Give each person a limited number of downvotes, say one for every five songs played. Downvotes are expended even if the song isn’t skipped.