Received 1099 MISC with an entry in Box 3 for a sweepstakes. Do I need to file in a state return where the Payer is registered?

No, you do not need to file for the State of the payer registration if you never lived or work there.

You can enter the 1099-Misc in TurboTax using the following steps:

1. Click on Federal Taxes > Wages & Income. [In TT Home & Biz:  Personal > Personal Income > I'll choose what I work on.]
2. Scroll down to the 1099-Misc and Other Common Income section. 
3. Click on the Start/Revisit box next to Income from Form 1099-MISC.  (See Screenshot #1, below.)
4. On the screen Did you get a 1099-Misc?  click on the Yes button. (Screenshot #2)
5. Enter the information from the 1099-MISC.
6. On the screen Does one of these uncommon situations apply? mark that This was prize winnings and click Contnue.  (Screenshot #3)
   
7. Continue through the rest of the interview screens.

The income will appear on line 21 of form 1040 (Other Income).