You won't have to worry about potential eye damage with a Full Moon or Blood Moon. This is due to it being a reflection of the Sun's light, which doesn't have any significant amount of UV radiation. As such, it's safe to stare into the full moon without fear of suffering permanent damage.
One thing to note is that the brightness of a Full Moon can cause the user to have dazzled eyes, which is a short-term side effect for long durations.
Therefore we recommend that a neutral density moon filter is used to reduce the brightness and lower the chances of feeling dazzled.