Try:
- iOS: Not responding or does not turn on
- Also try DFU mode after try recovery mode
How to put iPod touch / iPhone into DFU mode « Karthik's scribblings
- If not successful and you can't fully turn the iOS device fully off, let the battery fully drain. After charging for an least an hour try the above again.
- Try another cable
- Try on another computer
- If still not successful that usually indicates a hardware problem and an appointment at the Genius Bar of an Apple store is in order.
Apple Retail Store - Genius Bar
Replace/Exchange
Apple will exchange your iPod for a refurbished one for $199 for a 64 GB 4G and $99 for the other 4Gs They do not fix yours.
Apple - iPod Repair price
A third-party place like the following maybe less. Google for more.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Replace the screen yourself if you are up to it. However, more than the screen maybe bad
iPod Touch Repair – iFixit