http://www.youtube.com/watch?v=3K-16OIjnYA
The severity of problem appears to depend on the color and ink saturation of the tattoo, with darker, solid tattoos causing the most interference with the Apple Watch's sensors. A temporary solution for affected users is to toggle off Wrist Detection in the Apple Watch app under the General menu on the main settings tab, but doing so will disable Apple Pay from being used on the Apple Watch.
iMore also points out that natural skin pigmentation does not have the same affect on the Apple Watch's sensors, so there is no reason to be worried if you have darker skin. But if you have tattoos, the best advice for now is to try out the Apple Watch yourself and rest assured that the device has a 14-day return policy. Apple has not provided comment on the matter, but we'll keep you posted if we hear anything.