According to the great biographer and art historian Giorgio Vasari's magnum opus Lives of the Most Excellent Painters, Sculptors, and Architects (more commonly known as The Lives of the Artists), yes, these artists had an "extreme dislike for each other." However, it is never mentioned why, so we must result to speculation in order to understand their relationship. It is important to note that Leonardo was about twenty-three years older than Michelangelo, and was already a highly regarded artist in Italy by the time Michelangelo was making his start. One of the reasons I've heard the most was that Leonardo detested Michelangelo as he saw him as a "disrespectful youth," while Michelangelo disliked Leonardo because, well, he didn't really like anyone, especially artists with whom he had to compete with. One legend I've heard was that Michelangelo and Leonardo got into an extremely heated bidding war for the block of marble which would eventually form Michelangelo'a David, with Michelangelo ultimately winning out in the end, only further heightening the tension between the two. However, I've only ever heard this as legend and have never found an academic source to back this up.

Sources:

Giorgio Vasari, Lives of the Most Excellent Painters, Sculptors, and Architects (The Lives of the Artists). Ross King, Michelangelo and the Pope's Ceiling. Ross King, Leonardo and the Last Supper.

Edit: formatting Edit #2: Thanks to /u/The_Original_Gronkie for adding more information below!