There are several reasons as to why the teaching of the apostles would have created problems. For one, they were not priests and thus should not have been teaching the Jewish people, certainly not without authority from the council.1 Additionally, their teaching proclaimed that Jesus was the Christ, an idea which the Jewish council vehemently rejected when they had him crucified. The apostles are thus implying that the Jewish council made a mistake. Finally their objection was also due to the very idea of resurrection. The Sadducees in particular would be offended by such claims, viewing the apostles as a threat to their authority and position with the Roman overlords.2
2 greatly annoyed because they were teaching the people and proclaiming in Jesus the resurrection from the dead.