Some breeds of dogs and cats are known for their excessive tear production, salivation and a habit of licking their feet. The glands that feed the tear and saliva glands contain nutrients that come from the blood. The blood circulates around the body and carries these nutrients to all parts of the body including the glands mentioned above.
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When the tears and or saliva mix with the air, the iron in the blood causes the gland secretions to be reddish in color. Some animals produce color that is brighter red than others - it's a question of genetics, not a problem that a new diet can change. In fact, there are no studies that link these stains to any food ingredient or any food color. In fact, after a food is digested, after the nutrients enter the bloodstream, the original color of ingredients is long gone.  

The good news is that it doesn't signal a health problem.