If you don't use the 110 energy today, do you get an additional 110 energy tomorrow?

No. The timer refill will only fill to your bar's maximum then stops (110 for the original energy bar, 150 for the extended energy bar). For instance, you have the 110 bar and play until you have 0 energy. Then you open a star box and you win a bonus 50 energy. But you are too tired to play more and go to bed with 50 energy left in the bar. The next day you will find not 160, but only 110 energy, the amount you usually get in the morning. Timer refilled 50 to 110 then stopped. (So never open a box if you are not ready to play the energy that comes with it.)

However; you can add on top of a full bar by adding energy from other sources: star boxes, puzzles, coffee energy, landmark energy. In fact, that is how players stockpile huge amounts of energy for the competitions.