Continuity mistake: Shortly after Kong realizes he's been shot and gets blood on his fingers by touching the bloody hole in his chest, in subsequent scenes his chest has no puckered indentation and is unblemished but for minimal straight-line scars.
Continuity mistake: When Carl falls at the beginning of the brontosaurs chase we see that he is only just in front of the brontosaurs so when he fell he should have been crushed straight away but he isn't. Jack then runs to help him and we see that the brontosaurs only need another step or 2 (which would take 1 or 2 seconds) before they reach Carl and Jack. Carl gets up and argues with Jack for 5 seconds or so by which time they should have been crushed. They then run and there is a far off shot showing the brontosaurs back where they were at the start again.
Continuity mistake: Ann is soaking wet when she is being led through the natives. When she gets to the top of the wall she's dry.
Continuity mistake: When the Skull Island natives are holding Ann's arms as the old native taunts her, their grip on Ann's arms and sleeves differ between front and back shots.
Continuity mistake: When Carl and his assistant are driving in the cab discussing who to fill in on the female role, you can see that they pass a tram on their left side. When it cuts the cab is right in front of the tram rather than to its right.
Continuity mistake: When Carl is convincing Ann inside the diner, after she stands up and Carl convinces her to accept the role, there's a waitress behind Carl about to walk towards her. A split second later, while Carl says "she's the saddest girl he's ever met", the waitress appears 12 feet away, and is now behind Ann way at the end of the diner.
Continuity mistake: Kong continues to change size during the movie. The most obvious way to notice this is when his hands are near Ann. When Kong first finds Ann in NY, his hands are 3/4 the size they are immediately after when he picks her up.
Continuity mistake: When it cuts to the filming of the first take between Ann and Bruce on the ship, it shows Jack standing on the right side of the picture, about a feet away from the wall of the ship. When it cuts to a close-up of Jack, he is suddenly leaning against the wall.
Continuity mistake: When the "headsman" native is shot by Captain Englehorn, he throws his arms in the air. He repeats this gesture with the second shot as well. (01:01:20)
Continuity mistake: When Kong is about to fight the last Rex, Ann begins to back under his arms. In a distance shot we see that she is already a fair distance behind his arms, underneath him. Cut to Kong noticing this, then cut back to a back shot of Ann backing under and she has not even reached his arms yet.
Continuity mistake: When Carl speaks with Hayes and Lumpy about their destination, while aboard the Venture, the towel rags hanging on a line behind Carl change between shots.
Continuity mistake: Before going on board the ship, Captain Englehorn asks Ann if she is nervous while he lights his cigarette. You can see his cigarette remains in his mouth, but when it cuts it's no longer in his mouth.
Continuity mistake: When Kong is at the top of the Empire State building we see Ann start to climb the ladder and she is only a few rungs up when the camera view changes. When it cuts back to her, she is still climbing those first few rungs.
Continuity mistake: In a scene at the very top of the Empire State Building, Ann hugs Kong's left arm. In the very next shot, she is standing right in front of Kong, between both of his arms.
Continuity mistake: When Ann and Jack are about to kiss the first time, there is a close-up of Ann's face. When it cuts Jack's hand suddenly appears on her face.
Continuity mistake: After Ann and Carl arrive at the docks, as she stands beside the Venture her coat collar is flipped up, then turned down and back up between shots.
Continuity mistake: When Carl steps out of the room after screening his movie he doesn't cast any shadow on the door behind him. When it cuts his shadow suddenly appear on the door.
Continuity mistake: In the scene where Jack Black and Adrian Brody are talking on the boat (as it's getting ready to leave), when the camera is focused on Jack, the cityscape behind him (through the window) is moving (as if the boat has started moving), but when it focuses on Adrian, the cityscape behind him (through the window) is not moving.
Continuity mistake: When Ann is about to steal the apple, she picks up a different apple first. When she is about to put it back, the angle of the apple in her hand changes. Watch the apple stalk and you can see it.
Continuity mistake: When Ann is dancing in New York while King Kong is enchained, her hair is straight and then curly in the ends. But when she goes out in the streets and sees the Kong poster her hair is wavy. When she finally meets Kong again her hair is back to straight and curly.
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.