I would have a background process continuously doing .md -> .html conversion.
Then I would have a FireFox tab open on the .html, and using an add-on, set that tab to auto-refresh periodically.
Actually, regarding the first point, I wouldn't have a background process. In my Vim environment, I already have a custom command :W which saves the buffer and runs :make. I would have a Makefile which builds the .html from the .md.
So by saving the .md with :W, the .html would regenerate and the auto-refreshing FF tab would pick that up however many seconds later.