Alright. I played the game for a bit tonight and relived some awesome memories from my youth. I don't know if there's much better way to explain this than to draw a diagram, so that's what I did.
You start at the southeast corner and have to make your way to the northwest corner. Along the way are four green dots on the carpet. I've marked those with an X on the diagram. Each time you step on a green dot, the crazy furniture will run away from you. When you step off the green dot, the furniture will run toward you. Sometimes this can be quirky, since the furniture starts at a somewhat random location each time you respawn, and therefore each attempt will be a little different. But if you're having trouble making it to the next dot, take a step off of your green dot in the wrong direction from where you want to end up, and wait for the furniture to approach you, then go back to the same green dot. If you time this right, the furniture will run off to a different area of the room and there won't be anything between you and the next green dot.
So if you're having trouble getting past the third dot, try this. The furniture is usually to the west and north of you when you get to the third dot. You want to get all of the furniture northeast of you before you move toward the fourth dot. So once you've made it to the third dot, move one square south and wait a sec. The furniture should move toward you from the left, and if you wait until it's just about north of you and then move quickly back to the third dot, the furniture should move off to the north or to the east if you've timed things right. Then the path to the fourth dot should be clear. And once you make it there, it's a quick move off of the carpet and onto the moving tile part of the level. But in order to make it off of the carpet, you still have to have your timing right. The next portion has black bottomless pits with bluish/silvery tiles moving back and forth, one square at a time. You can't move off of the carpet until the first square next to the carpet is bluish/silver. Once you make it that far, it should be clear how to move on.
I hope that helps. I don't mean to analyze things in too much detail, but I don't know any other way to explain exactly what to do in that part of the game. Let me know if you're still having trouble with it.