Can I do online work (remote work for a foreign company) while on a tourist visa in Korea?

Working remotely for a foreign employer while physically in Korea on a tourist visa is a legal gray area and technically not permitted as paid activity, even though people do it short-term. If you want to stay longer specifically to work remotely, look into the digital nomad visa (F-1-D), which is designed for exactly this. Definitely don't take on locally paid Korean work on a tourist visa, as that's clearly not allowed.