I've worked with a lot of illegal immigrants (a lot, a lot) and in my experience, they do pay their taxes, even if only because taxes are taken out automatically from their checks. The real issue is lack of documentation for workers, be they "legal" or "illegal". White, native, legal citizens of the US get away with paying taxes all the time; I know people who own lawn-care services and pay nothing in taxes. The point is that tax law isn't being enforced very well. I don't believe in taxation, so my political ideology bypasses the issue altogether. I'm for open borders.
Unfortunately, no, illegal immigrants are not citizens of the United States and there are no requirements for them to file taxes, because they are not even suppose to be here. This is one of the major problems with immigration reform, because illegal immigrants, in some cases, have used governemnt programs for food and assistance, yet, they don't fund these programs.
No, a fair share of taxes are not being paid by immigrants who are in the United States illegally, because they do not have to file tax returns. Immigrants can also list a large number of dependents, if they do file taxes, and there is no way to check, so they end up receiving large refunds that they do not deserve.