What happens if a fan runs onto the field and tackles a player during a game of football?

The person is taken into custody by security personnel and turned over to the police who have jurisdiction. There are several charges that could apply from misdemeanors such as of creating a public nuisance/disturbance to felonies such as assault and/or battery (the player would not have to be injured for an assault charge to be valid. -----
As to where the ball would be placed, that would be upto the discretion of the officials. There have been instances in College Football history where players on the bench have tackled ball carriers that were running for touchdowns. The officials ruled those plays as touchdowns. The person would most likely also be crushed super hard by one or more players on the field before security could get there. Some people are just stupid like that.