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 intoBottomless Pits trying to get you. Heck, even fleeing civilians and Tyris' magic flame will fall into them. Also, the most effective way to beat Duel is to get 2 enemies on opposite ends of the screen and keep fly kicking off them, like a pendulum swinging left and right. Mario enemies are smarter than this.
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.
Nickelodeon's Teenage Mutant Ninja Turtles has downright idiotic AI for your allies. The developers obviously figured out that the game was way too easy even in single player, so your allies will do absolutely nothing to help you. One Turtle will fight one enemy (launching attacks so slowly that you'll probably have to finish the enemy yourself) and the remaining AI turtles will stand around doing squat. Enemies will sometimes join them, standing around and waiting until you're not busy so that you don't get dogpiled with too many enemies at once, yet not bothering with the charade of pretending to fight your allies. The AI turtles also run much slower than you and will vanish off-screen if you run too far into a level, preventing you from switching to them until they respawn or until you backtrack and find them. They also don't help you during bosses, instead opting to stand around getting clobbered by the boss again and again. Thankfully they take much less damage than you, or they'd be deep in Stop Helping Me! territory.