Does the word "luve" mean "love"?
The short answer: Yes.
Standardized spelling is something very new to the English language, as strange as that may seem to us. Go back just two hundred years or so and you'll see people spelling words all sorts of ways. There wasn't the standardized educational system that we have today, and there were far fewer dictionaries and other sources that we now turn to when we want to check the "correct" spelling of a word.
Robert Burns, the author of the poem you name, lived from 1759-1796, toward the end of the stage of English known as the Early Modern Period. By this time, a standardized spelling had pretty much already been developed and disseminated, at least to the middle and upper classes, but it wasn't identical to our standardized spellng today.
Here are some words from his poem paired with a modern (American English) spelling:
luve = love
melodie = melody
weel = well
On a final note, I don't have much proof to support my claim, but I believe that Burns did not write his poem in the way that he actually would have spoken in everyday life. He uses the word "thou" in the poem, for example. The word "thou" had already pretty much been entirely displaced by "you" and become a fossil by the time he was writing this poem.