If I trap the heat on the hot side of Peltier using a vacuum flask where no heat can escape to the surroundings so that the room can get cooled faster. Would it reduces my Peltier's efficiency and damage it? Or must I remove the heat as much as possible from the hot side to improve its efficiency?

 

1) Your Peltier unit is at best 3-5% efficient, it will always generate far more waste heat than its cooling power.

2) If you do not remove the heat from the hot side, it will overheat and fail.

3) Heat flows from hot to cold- if you tried to "bottle it" - the best you could hope for is that the internal temp of your bottle would match the heat source.

 

If I trap the heat on the hot side of Peltier using a vacuum flask where no heat can escape to the surroundings so that the room can get cooled faster.

Why would you think that would help?
The Peltier removes heat from the cold side and moves it to the hot side (like an air conditioner).
The hotter the hot side gets, the poorer the transfer efficiency of the Peltier.

 

How does one trap heat?
What is heat?

Heat is thermal energy.
Where is this energy? Thermal energy is represented in the motion of atoms and molecules. You can transfer this energy via three mechanisms, conduction, convection, and radiation.

The efficiency of a Peltier device or TEC (thermo electric cooler) is dependent on the temperature difference between the hot side and the cold side. The heat energy on the hot side must be removed otherwise the temperature will rise and the device will be destroyed.

You remove the heat using conduction, convection, and radiation. You do not want to trap the heat.

Imagine that the TEC was mounted to an almost infinitely large heat sink or water tank.
The thermal capacity of the tank is so large that it will maintain the temperature of the hot side at the given ambient temperature. Furthermore, the tank itself does not rise in temperature and neither does the room.

This will work to cool the air in the room only if the thermal capacity of the air is lower than that of the tank. The overall net result is the total energy in the room rises because power has to be applied to the TEC to operate it.

 

There are three laws of thermodynamics:

1. You cannot win
2. You cannot break even
3. You cannot get out of the game

 

MrChips' explanation is correct. To highlight, once you run a fixed current through a peltier, it attempts to maintain a fixed temperature difference between the hot side and the cold side. If you "bottle" the heat, the temperature inside the bottle will rise until it reaches the cold side temperature plus that temperature difference. When this happens it will no longer cool the cold side. Given the inefficiency of the peltier it will overheat and destroy itself.

It actually might work if you had a large enough bottle (maybe a few cubic meters in volume). However a freon based cooling system would be far more efficient at that size. There are heating and cooling systems that store heat for later use or for heating tap water.