No, not unless the particular clothes you choose that day cover your wrists. Tattoos on guests is very normal. Many many guests have single tattoos, full sleeves or multiple back , arm or leg tattoos with all sorts of Disney themes and designs. You will also see guests with face tattoos and all sorts of non Disney tattoos.
I have 5 large black tattoos, 2 at the top of one arm, one on the top of the other arm and one on each shoulder blade on my back.
The only thing that concerns me when in Disney parks is the sun. I am from Ireland and my skin is not used to the strong California sun, so I always have a cardigan or hoodie or wrap to protect me from the sun.