Not when the tattoo is healed already
In fact, you only need to put A&D or Aquaphor on a tattoo for the first 3 or 4 days, to act as a second skin. After that the scab is formed and then you really are just conditioning the skin. I personally use CVS Cocoa Butter, but any moisturizing lotion is good.
But red definitely can cause a reaction, which will itch like a mofo well after the tattoo is healed.