Continuity mistake: In the scene where the boat is leaving NYC and the camera is panning around it - the smoke from the ship "follows" the camera by always pointing to the right relative to the camera view; even though it should be trailing backwards, opposite of the ship's direction.
Revealing mistake: As Kong and Anne Darrow exhale in the freezing New York air there are many shots where you can't see their breath frosting up as it comes out of their mouths.
Continuity mistake: After the screening of Carl's movie he walks down and says "The script has been re-written", while he has his left hand up by his waist. When it cuts his left hand is hanging down.
Continuity mistake: While telling Carl about the shipwreck sailor the crew of the venture picked up on an earlier cruise, Lumpy keeps whittling on a piece of scrimshaw. He continues to do so when Carl is about to leave the galley. In the next shot, when Carl turns back to give a reply, Lumpy's arms are suddenly lowered at his side.
Continuity mistake: During the whole film, even though Ann is thrown about and man-handled by Kong, as well as sliding down rock and cliffs in bare feet, and falling over numerous times to escape the many creatures and dinosaurs, she ends up with virtually no marks or scratches on her body. Her dress is dirty, but that is about it. Her hair also stays reasonably in place. Surely she would have been scratched and bruised all over her legs and arms from the trauma.
Continuity mistake: In the scene where the movie star is trying to talk them into giving up he has a rope slung around his chest over one shoulder. He never takes it off, but when he leaves he no longer has it, and Adrian Brody is holding it.
Continuity mistake: When the main characters first arrive on the island and are attacked by the natives, one of the natives attempts to decapitate Carl but Englehorn arrives at the last minute and saves him by shooting the native three times. As he is falling, it is very noticeable that there is not a scratch on him. He never bleeds at all, there is not even an entrance hole for any of the bullets.
Continuity mistake: When Kong has grabbed Hayes, he tells the rest of them to get out of there. As he says "Do as I say", his gun is pointing at Kong. When it cuts he suddenly has his other hand on the gun cocking it.
Factual error: In the scene in which Kong is running loose on the streets of New York city, and being chased by two military Army vehicles, the vehicle identification number, starting "USA and followed by six or seven numbers" is visible on the hood of the vehicles. The first two numbers in a USA number is the year of manufacture of the vehicle. Thus, the truck with the 3 inch cannon was manufactured in 1945 because the number starts "USA45." The time setting of the movie is 1933-1934.
Continuity mistake: When the old man says he is moving back to Chicago, Ann hugs him and place her hand around him right below his neck. When it cuts her hand is lower down on his back.
Continuity mistake: After the running with the dinosaurs scene. Driscoll and Baxter are talking near a swamp. In the first shots of Baxter, he has a coil of rope over his shoulder. We see a shot of Driscoll and when it goes back to a shot of Baxter, the rope is gone. We never see Baxter remove it or see the rope laid down as he walks away. (01:28:50 - 01:29:20)
Revealing mistake: When Kong is shaking the tree down it's obvious that there's a screen behind the film crew when they fall.
Continuity mistake: Aboard the Venture, when Ann runs into Jack with her "good legs" she has a towel on her shoulder, which changes position until she goes to her cabin.
Continuity mistake: When Ann is caught stealing the apple, Carl pulls up a coin saying "Excuse me. Ma'am, I think you dropped this". The way he holds the coin differs in the next shot.
Visible crew/equipment: As one of the Venture crew members picks up a chloroform bottle from a case, you can see a bright studio spotlight reflected in the bottle.
Continuity mistake: When the sound engineer says "We're gonna have the ship's engines all over the dialogues", he has his left hand to his headphones. When it cuts his hand is gone, but in the next shot his hand is back on the headphones.
Continuity mistake: Jack and Carl are in Carl's stateroom. Carl opens the crate of whiskey, takes out a bottle and puts it down. The bottle then turns itself around so the label's sometimes visible, sometimes not.
Plot hole: In the scene where Ann is laying/sleeping with Kong on the cliff, her dress should not only be beyond filthy, but it should be tattered and shredded. It doesn't so much as have a thread out of place after being carried so violently through the jungle and being caught between Kong and the dinosaur. Also, she at this point has been in Kong's possession for several hours to a day or so, and not once has she even shown any signs of hunger, thirst, or the need to relieve herself.
Continuity mistake: In the scene where Ann is unconscious, Kong puts Ann on the ground and she is lying on her arms. When it cuts to the shot when he begins poking her, one of her arms is stretched out.
Continuity mistake: When Ann and Kong are on top of the Empire State Building near the very end, he sets her down on top as he grows weaker. She begins to stand, pulling herself up on his fingers, the shot cuts and she is standing but his hand is now out of place back towards his body, not near Ann where it should be.
Answer: I assume you mean, could the building take his weight, not whether an ape would really have the ability to climb a building (if that's what you mean, then it's definitely yes...apes are great climbers). Assuming Kong is proportionally as heavy as normal-sized gorillas, which tend to be in the area of 160kg (~350lbs), then he weighs over 80,000kg (89 tons, give or take). The average human weighs about 62kg, so that's about 1,300 humans, and the capacity of the ESB is over 13,000. So, assuming the building is mostly, or even half, empty while a giant gorilla scales it, the building could handle his weight.
Keep in mind, though, that the weight allowance for the building assumes people on the floors of the building, not climbing on the outside. The outer structure of a building isn't designed for massive creatures climbing on it. While the building as a whole would likely survive, there would be significant damage as Kong would be breaking windows and pulling stone off it as he made his way up.