Be careful not to mix up tones and intonation.
A language with tones will distinguish words with different pitches or pitch contours - this means that you can have two words with the same phonemes, but which are distinguished by pitch (or pitch contour -rising pitch, falling pitch, etc.) alone. Chinese has 4 tones, because, for example, the word "ma" can have 4 meanings based on the tone: (mā 'mother', má 'hemp', mǎ 'horse'and mà 'scold').
Intonation, on the other hand, means changes in pitch or pitch contour that doesn't change the meaning of words, but can change or enforce the meaning of the sentence as a whole - for example, the rising intonation used in English to indicate a question.
Standard Korean does not have tones, though they did exist in Middle Korean. The Gyeongsang dialects also have tones. In the case of 눈 that you mentioned, tone is not involved at all - only vowel length.
There is one way in which pitch is important in Korean. After the tense consonants (ㄲ,ㄸ,ㅃ,ㅆ,ㅉ), you will usually notice that the vowel that follows is higher-pitched than other vowels. This technically does not count as lexical tone1, and it isn't necessary to pronounce these vowels higher-pitched to indicate the meaning - but it is easier to distinguish the tense consonants from the non-tense ones when you consider the pitch of the following vowel.
Intonation is important in Korean as in all languages, but it differs widely according to dialect. In Seoul Korean, the pattern of rising intonation for questions and falling intonation for statements holds in general.
Finally, if you want a really in-depth look at intonation and tone in Korean (dialects included), here's a dissertation about it.
1 Technically, lexical tone means that it acts to distinguish minimal pairs - that is, two words can only be distinguished by the tone. With Korean words like 곳 and 꽃, the two words are distinguished by the first consonant - the higher-pitched vowel in 꽃 is incident, and isn't even necessary.