The biggest may be the way that it displaces native students who've often been paying big taxes over the years to improve the schools. Yet they get bumped to a lower tier school or forced to pay for private education.
It's one thing when the incoming talent is undeniable, but in many cases schools will actively (and openly!) search for foreign students who pay substantially more in tuition.
There are also problems with language for some incoming students. I know of at least one state school that is actively turning away state residents while scolding the professors for grading foreign students for poor grammar and English writing.