If he's feeling sick, then there's nothing you really can do. If he's teething too. You can try ignoring it, but I've known grown men who whine when they feel ill.
One of the few things I know that works is teaching different voices. "Let's use our whiner voices..." "let's use our inside" "let's use our happy" "LETS USE OUR OUTSIDE VOICES" etc. Then, when they start whining, you say "Oh, want to use whiny voices?" and whine. Then say something like "remember to use your happy voice when asking for something."