I recently purchased a used Phantom 3 and the guy lost his receipt and paid in cash. If my phantom fails due to firmware upgrade, would DJI fix my phantom for free? I'm not talking about crashes...just issues caused by the upgrade failures like no video transmission after the firmware upgrade.

If upgrade failed to perform then you just roll back to previous build.

Without a receipt for the original purchase, DJI will probably not look at your warranty.

As Geebax stated, warranty service is determined by the proof of purchase. Without a receipt or a DJI order number, you would have to pay for any potential repairs.