I'm going to go through a list of all moves in Bulbapedia which don't make sense to me.
Razor Wind/Whirlwind - not flying?
Mist - how does water vapor shield you from Screech/etc?
Agility/Rest - how psychic?
Mind Reader/Psych Up/Disable- how are these
not psychic?
Teleport - how does Fly allow you to go anywhere, but not this?
Smokescreen - how does this only hit one opponent in doubles?
Bone Club/etc - how do these miss flying types? Mud attacks, too.
Curse - how does this japanese pun for "slow" also raise your atk/def?
Milk Drink - Eww.
Sleep Talk - shouldn't it be Sleep
Walk?
Rapid Spin - why should the defensive properties fail against ghosts? The leechseed/hazards are on
you.
Moonlight - stronger in the sun?!
Dive/Dig/Surf/EQ/Rock Climb - should fail, depending on the environment
Gastro Acid - and vomiting stomach acid gets rid of abilities
how?
Wring Out - how is this a special move?
Grass Knot - how does this make contact when things like Starmie use it?!
Heart Swap - in-game it swaps stat boosts. In the anime/brawl, it literally switches souls.
Magnet Rise - specifically when a magnemite has to use it
Mirror Shot/Flash Cannon - how does the light reflecting off these surfaces crush rocks, etc?
Power Gem - same as above
Wide Guard - is a rock
Soak - how does submerging a Charizard in water turn it into a water-type without taking damage?
Clear Smog - references water vapor that comes out of chemical plants, yet is poison
I bolded the ones that I think are the least logical, for your TL;DRing pleasure.
COTTON GUARD AND COTTON SPORE
Seriously GameFreak? How the hell does covering yourself in cotton balls increase your ability to withstand Flamethrower. And how does it lower your speed??? WTF?
Cotton Guard raises physical defense, not special defense. It makes perfect sense that if your opponent tackled you at full force or threw something heavy on you, that a cushion of fluff would block the damage. Cotton Spore doesn't make sense, though.
String Shot lowering speed? It's string, the pokemon could just pull it off
It's not string, it's comparable to the material used in spider webs. A spider the size of Ariados should easily be able to trap and restrict the movement of its foes.
pretty much all water type moves are unrealistic. hydro pump could work, but water gun and bubble would be a mild irritant. anime comes to the rescue by making the bubbles explode, but still.
also I love how the existence of quick attack means that, in realistic terms, pokemon that use tackle just aren't trying very hard
Imagine water gun being launched at the pressure of a firefighter's hose, like it is in the anime. Then imagine Hydro Pump being 3 times the PSI of that.
Most pokemon that learn Quick Attack learn it after Tackle. That's like saying that pokemon that learn Ember aren't trying hard enough because they're not using Flamethrower.
Scald. Just imagining hot water catching something on fire is hard for me to grasp.
I personally do not interpret BURN status as being lit on fire, but rather having burned, damaged skin, such as a sunburn. If you were to dip your hand in boiling hot water, it would be hot enough to sting for a while and would hurt to move (i.e, lowering your attack)