If a character becomes exhausted from card loss, all of their cards are removed from the game, as per the normal exhaustion rules. These include all summons and active cards.

They don't get a turn per se but are treated as having one for the effects and expiration of conditions. The processing of that turn is nominally at initiative 99 and takes place after long rests. Existing conditions are removed at the end of that turn per the condition rules.

The character may still be focused by enemies per the normal focus rules (including activation order tie breaker rules). They can still be affected/targeted by friendly and hostile abilities, including receiving new conditions. Some conditions won't be relevant, like bless or strengthen, but invisibility would be. They may be healed to keep them alive in case of future damage but remain unconscious for the remainder of the scenario even if healed.