Hi, Shane! If Walt Disney World has to close a theme park due to capacity, there isn't anyone who would be permitted through the turnstiles. A sufficient number of guests would have to exit in order for them to allow anyone else in, and it doesn't matter if you have a dining reservation inside the park. Once the park is full, it's full.
I will tell you that the park closes in stages, and on-site guests and Annual Pass Holders are the last groups to be affected by park closures. You may have SOME advance notice that the park is filling up by checking with the concierge at your resort.
Your best bet really is to just plan to spend your day at the Magic Kingdom. New Year's Eve is one of the busiest days at Walt Disney World, and with such a coveted dining reservation on one of the busiest nights of the year, it would be such a shame to miss it! You can plan your day so that you do the bulk of the touring around early, then plan to spend the afternoon with a nice lunch, enjoying a parade, or doing things that will help you relax and enjoy the afternoon while you wait for your dining reservation time.
You couldn't have chosen a more magical time to visit Walt Disney World! I hope your vacation is spectacular!