Inspired by this question about Frightened.
According to Appendix A: Conditions (SRD p. 192),
A condition lasts either until it is countered (the prone condition is countered by standing up, for example) or for a duration specified by the effect that imposed the condition.
While a character is Unconscious (SRD p. 193) it
- Can't move or speak, and is unaware of its surroundings
- Automatically fails Strength and Dexterity saving throws
If a PC under the effect of a persistent effect (such as the Sea Hag's Horrific Appearance) that allows for a save to be made each turn and then becomes unconscious, does the character continue to save against the effect?