Nothing. But a population that is already unlikely to cooperate with the government is even less likely to cooperate if they know a government official will be asking them about their citizenship status. That results in an under count of the population, which affects things like how many seats in the house (or the number of college electors) a state gets apportioned.
Because illegal immigrants tend to live in Democratic districts, even inside red states, this has become a partisan issue. While illegals cannot vote, citizens living in the same district essentially get more voting power (there are ~750,000 people per US house representative, but only the citizens inside that district get to vote for the representative).
That's why I think it makes sense to go back to counting illegals as 3/5 a person, but these days I bet that would go over like a fart in church.