The bottle contains a few drops of water. A wet coin is placed on the bottle's mouth (orifice). The coin jumps (a bit) up and down a few times when hands are placed around the bottle.
Question: When the bottle is held with the hands (before the coin jumped),
Solution: The air inside the bottle is cool. The temperature of the hands (approximately 37 °C) is more than the temperature of the air inside the bottle (Sometimes hands are vigorously rubbed before putting them around the bottle.). Thus, the heat gets trasnferred from the hands to the inside air through the bottle's wall.
The coin is made wet before putting it on the bottle's mouth (orifice). The wet coin separates the inside air from the outside air. The inside air cannot move out without lifting the coin.
The volume $V$ of the bottle (and hence air inside it) is constant. This is ensured by putting the hands around the bottle without pressing the bottle.
The heat transferred from the hands to the air raises its temperature. The rise in air temperature increases air pressure ($pV=nRT$, at constant $n$ and $V$). The increase in pressures increases the upward force on the coin. The upward force lifts the coin up that allows air to move out.
The pressure inside the bottle drops when air moves out (due to decrease in $n$). This decreases upwards force on the coin and the coin comes back to its initial position. This process repeats many times.
This demo is a simple form of a heat engine. You can visualize the work done by the heat.