I was reading an article in the New York Times that explained a lot about illegals paying taxes. I will find the whole article later but here's what they said.
Yes, illegal immigrants pay taxes. What percantage of them that do is impossible to figure out. The reason why they aren't getting anything back is another topic entirely. They are buying fake id's and fake greencards that include fake social security numbers. When the government goes to give refunds and the like they have all these w-2 with fake numbers on them, so who should they give them to?
Edit: Found one of the articles:
The New York Times > Business > Illegal Immigrants Are Bolstering Social Security With Billions