Will my laptop get damaged from being too hot?
Yes, heat damages any electronic device, including laptops. When they get hot, they will also run slower.
Laptops usually have their intake air vents on the bottom of the computer and they blow the air out normally on the sides.
This design presents a problem in terms of where laptop computers are usually used and that is on a laptop which has normally got your clothes between you and the laptop. The clothes are flexible and often will block the air coming in the computer and thereby the air is not flowing through and cooling the various parts of the computer that are getting warm and thereby they get hot. Placing the laptop computer on the bare skin of your legs can cause burns from the heat but usually the laptop is drawing enough air to dissipate heat as usually our skin does not fully block openings on bottom of laptop.
With the heat you can literally burn parts up and damage the parts.
Using the computer laying on a bed on the bed covers or laying it on carpet while you use it laying or sitting on the floor will present similar problems.
Two ways that work well around these problems are the following:
First put a hard surface like a board or a student hard surface study board between the computer and the soft fabric. Now the air can flow under and up thru the computer. The other is use the laptop computer on a desk or a table, but then it is not being used as name but many, including me, use this way.
The second is that they sell cooling platforms that you set the laptop computer on and you plug that cooling panel in the USB port on your laptop and there is one or two fans in the cooling panel that force air in the laptop. Not only does this prevent the laptop form over heating but it will keep it cooler and often then it runs faster.
The cooler you can keep your computer the better it will run. Setting the laptop on clothes on bed covers or carpet can potentially be a fire hazard also.