In my experience, it is always been the ratio of products to reactants that mattered, so as long as your units are consistent; what they are doesn't matter. However, an in-class problem today required that the units be changed to molar (M) from millimolar (mM). Can you summarize when/why this matters in some cases, but not in others?
$K$ should be a unitless value. Otherwise, you'd get a nonsensical unit when you try to take the log of the unit. Each component of the product $K$ is usually an activity, which is frequently similar in value to the molar concentration, or a fugacity, which is frequently similar in value to the partial pressure of a gas.