In the arcade version of the first Double Dragon, enemy characters tend to backpedal when the player starts throwing punches at their direction and they will only approach when the player has his back turned on them. At first, this seems like a sound strategy to employ until you realize that you can attack enemies with the elbow strike move (by pressing Punch and Jump simultaneously). The elbow strike is a ridiculously powerful move that has a decent range, does quite a bit of damage, and can knock enemies to the ground in one hit (the same as doing an uppercut or a roundhouse kick). The enemies will always fall for this trick no matter what. As a result, the entire game can be completed easily in one credit by using only this move. The arcade version of Double Dragon II smartened the A.I. a bit to make enemies less likely to fall for this trick, while the later GBA version nerfed the elbow strike to the point that it was no longer useful.
The Ninja Warriors Again has a stage where giant fans are spinning vertically from the ceiling. In that area, the enemies will repeatedly walk into the fan trying to approach you until the enemy kills itself from its own stupidity.
In Golden Axe, every enemy can fall into Bottomless Pits trying to get you. Heck, even fleeing civilians and Tyris' magic flame will fall into them.
In the Fan Remake of the Streets of Rage series, the AI partner sometimes refuses to heal itself when needed or to use a Police summon when in a pinch. It doesn't use very efficient attacks often and likes to break your combos if set on aggressive. The worst, game-breaking, situation that may happen is when the AI refuses to follow you into the stage. In the rooftop level, for example, you're forced to waste a continue to get it to jump over a pit.