In some states, the local party has to swear to the eligibility of their chosen candidate. Maybe this isn’t the case in Tennessee, but what would be the point in nominating someone who ultimately is NOT eligible for office unless that party is admitting they intend to break the law??
Perhaps Obama is ineligible, but Tennessee isn't going to try to find out. Perhaps they hope that Obama will get enough votes that no one will have courage at the federal level to say "Stop! You can't be president!" After all, that is what happened last time. Pelosi signed two pieces of paper, artfully declaring Obama was eligible. John Roberts muffed the oath of office then had a do-over in private. There is plenty of evidence that at the highest levels they know Obama is not eligible and they do not care, because they are afraid of race riots. So, as long as states like Tennessee don't ask too many questions about eligibility, he can be put on ballots, he can be nominated, and he will just be waved on through to a second term.