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Ok, lets break this down piece by piece. If you aren't interested skip to the end
Description: Rendering entity in world
This means that an entity in the world is causing a crash while being rendered. A common cause of this is that an entity from a mod in your world has an issue with the model.
java.lang.ArrayIndexOutOfBoundsException: 29
This means that an array in the render code contains less than 30 objects but the 30th object is trying to be accessed.
at net.minecraft.client.gui.MapItemRenderer.func_78319_a(SourceFile:44)
at net.minecraft.client.renderer.tileentity.RenderItemFrame.func_82402_b(SourceFile:150)
at net.minecraft.client.renderer.tileentity.RenderItemFrame.func_82404_a(SourceFile:44)
at net.minecraft.client.renderer.tileentity.RenderItemFrame.func_76986_a(SourceFile:20)
at net.minecraft.client.renderer.entity.RenderManager.func_78719_a(RenderManager.java:312)
at net.minecraft.client.renderer.entity.RenderManager.func_78720_a(RenderManager.java:281)
at net.minecraft.client.renderer.RenderGlobal.func_72713_a(RenderGlobal.java:524)
at net.minecraft.client.renderer.EntityRenderer.func_78471_a(EntityRenderer.java:1160)
at net.minecraft.client.renderer.EntityRenderer.func_78480_b(EntityRenderer.java:1006)
at net.minecraft.client.Minecraft.func_71411_J(Minecraft.java:946)
at net.minecraft.client.Minecraft.func_99999_d(Minecraft.java:838)
at net.minecraft.client.main.Main.main(SourceFile:101)
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(Unknown Source)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(Unknown Source)
at java.lang.reflect.Method.invoke(Unknown Source)
at net.minecraft.launchwrapper.Launch.launch(Launch.java:131)
at net.minecraft.launchwrapper.Launch.main(Launch.java:27)
This means that a map in a item frame is failing to render.
-- Entity being rendered --
Details:
Entity Type: ItemFrame (net.minecraft.entity.item.EntityItemFrame)
Entity ID: 57
Entity Name: entity.ItemFrame.name
Entity's Exact location: -239.50, 51.50, 83.06
Entity's Block location: World: (-240,51,83), Chunk: (at 0,3,3 in -15,5; contains blocks -240,0,80 to -225,255,95), Region: (-1,0; contains chunks -32,0 to -1,31, blocks -512,0,0 to -1,255,511)
Entity's Momentum: 0.00, 0.00, 0.00
This is the entity that is failing to render. Take note of the coodinates (highlighted), you'll need them in a second. That's really all that matters to solve this.
Now, on to fixing it
First of all, If you don't have it already, download MCEdit. Open it and click "Open a level..."
Navigate to your saves folder and open the world.
Hit ESC to open the menu, click "Goto", and type the coordinates of the entity (you wrote them down earlier) in the box.
x: -239, y: 51, z: 83
You should see a brown box. While looking at it, right-click and then double-click. Click "Delete Entities"
Finally, hit ESC, click "Save", and click "Quit"
Load the world up in Minecraft and it shouldn't crash.