I think it falls on the entire list.  But, I think it should be led by the conference, coupled with the host school.  The conference should mandate specific requirements for marketing and advertising as part of the host site package, but be fully prepared to support and assist, both financially and with staff.  I am unaware of the current process, but the entire conference either benefits or pays the price for poor tournament outcomes.  The host site puts a great deal on the line, but the conference itself should take the lead in making sure all tournaments are as absolutely successful as possible.  
Once the conference takes the lead, and the host site accelerates the process, it is the host site school's responsibility to advertise and excite the local community.  
One provision should be that any host site that does not make the tournament that they are hosting, all of the admin, coaching staffs, and players in all sports must walk the streets of the city, video-taping the process for the record, and draw the crowd.  They also must appear at the event, handing out candy to the children (sugar-free of course), and must perform at intermission of each game, doing skits and such (the more degrading the better).  Yep, life should be hell when you don't make your host tournament (more than just the standard humility that should exist).  That way, there is no way you will ever, ever let it happen again.