Less uniformly than its reputation suggests. A large part of the classic repertoire has no chilli in it at all.
Korean heat comes mostly from gochugaru (dried chilli flakes) and gochujang (fermented chilli paste). Gochujang is sweet and thick, so dishes built on it — tteokbokki, bibimbap sauce — taste sweet before they taste hot. Gochugaru heat is sharper and more direct, and it is what makes stews and kimchi bite.
Samgyeopsal and most grilled meat, bulgogi, galbi, japchae, samgyetang, kalguksu, gimbap, jjajangmyeon, tangsuyuk, mul-naengmyeon, sundubu in its white version, and almost all fried chicken except the yangnyeom kind. It is entirely possible to eat in Korea for a week without meeting chilli.
Buldak (fire chicken), jjamppong, yukgaejang, agu-jjim, nakji-bokkeum, and any dish with 매운 or 불 in its name. Kimchi-jjigae and tteokbokki sit in the middle — hot but manageable for most people.
'An maepge haejuseyo' (안 맵게 해주세요) means 'please make it not spicy'. It works reliably for dishes assembled to order — tteokbokki, stews, stir-fries — and not at all for anything pre-made or fermented, like kimchi. Milk or a soft drink helps more than water, which spreads the heat around.