I don't think there's anything wrong with removing tables. I work at a small banquet hall and we do this frequently. The tables are not removed until after coffee and dessert service is finished and cleared, so the scandalized people who are picturing guests eating their meal on their laps need not worry.
My suggestion is to call attention to an activity on the other side of the room. Where I'm from its common to serve a separate dessert, then cut the cake and serve the cake as a late night buffet.
The timing usually works out like this: tables are cleared, speeches end, mc announces the bar is open and the cake is being cut.
Odds are most people will either head to the washroom, bar, cake cutting or outside to smoke. Anyone left at the tables being removed is asked politely to pick up their belongings. 4 or 5 staff members swoop in and the tables are gone within 2 minutes. The chairs are added to existing tables. Everyone comes back in and the first dance is done and the guests barely notice the difference. After the first dance the dj starts up and most people are up and dancing anyway.
My two cents for what it's worth.
Talk to your coordinator but odds are the staff has done this many times before and can make some recommendations.
Well thanks again for your help we have decided to move the dj and dancing and photo booth to the pavilion right next to the clubhouse. We will serve cake and coffee after dinner then invite everyone over for our first dance. Untraditional yes but that is who we are, there is seating for everyone outside as inside seating will remain in there as well and I think it will work much better then removing tables so thanks again all.