Hi, I don’t know this answer as I’m new.
What would happen to the bees if for whatever reason you don’t harvest your honey and kept the frames full with no further room for storage. Naturally if there’s no room why forage and if they continued where would they store it? Would they start to use the brood box to do so?
Thanks.
Usually they will swarm if you do this, taking a lot of the honey and more than 60% of the bees with them. 
The brood will already have stores. The super is the back up. Bees are natural foragers. Honey bound hive with no space the bees will look to expand by naturally swarming.
Not the end of the world and nature just doing its thing.
The first place the bees will store honey is in the brood box so it is handy for feeding the brood. Then they will make use of the super. Bees can’t ‘turn off’ the urge to forage, they will make comb outside of the frames and then they will either swarm or even abscond completely.
Cheers