To show $2$ colors doesn't work, consider three numbers $2\cdot 19$, $2\cdot 53$, and $19\cdot 53$. By PHP, there are two of them having same color, whilst their product is a multiple of 2014. Then to show $3$ works, first color all multiple of $2$ that is relatively prime with $19$ by red, then all multiple of $19$ that is relatively prime with $53$ by green and then the rest by blue. It works because whenever $2014|ab$, exactly two of $\{ 2,19,53 \}$ must divide one of $\{ a,b \}$ and the other prime must divide the other number. But, these numbers have different color, so we are done.