I don't know about illegal immigrants but there used to be a thriving business for scammers and fraudsters. These days you're asked to go to one of the identity verification offices of the IRS with a state issued id and social security card.

This applies to previously flagged accounts, audited filings etc.

http://www.cnbc.com/2015/02/11/tax-refund-fraud-to-hit-21-bi...