Do you have to putt out on every hole in a bogey competition even when you are over par on a hole ?
No, it is played on a hole-by-hole basis so once you have lost the hole you can pick up. The rules even encourage you to pick up when you are out of a hole.
Scoring in Par/Bogey
If the player’s score is higher than the fixed score, or no score is returned, the player loses the hole.
A player who does not hole out under the Rules for any reason loses the hole.
To help pace of play, players are encouraged to stop playing a hole when their score exceeds the fixed score (as they have lost the hole).