All the common "rules" of what hot air does, etc. do NOT apply when you use FANS to force that air to flow somewhere. Sure, wherever it goes it carries the heat with it.
Inside a computer case the flow of AIR is how heat is moved away from hot components and ultimately out of the case into the room. Since the room air is what is brought into the case for this, heating of that air is a small factor. The impact on room air temperature is not huge, however. Consider that a heavily-used computer may consume 600 W power, so that's like having a 600 W electric room heater operating. If that were to warm the room so much you find it uncomfortable, you'd find a way to cool the room. BUT YOU would be "uncomfortable" at a much lower temperature than what might reduce computer innards cooling significantly.
It would be rare that the room air temperature is so high that you might want to cool it BEFORE it enters the computer case. For purposes of removing the heated air from inside the case to the outside by fans, there is absolutely NO need to cool it first!
A liquid cooling system (typically used for the hottest component, the CPU) is a means of removing heat from the CPU and getting it out of the case. Good ones ARE better at this job than the best simpler systems of a fan cooling a static heatsink mounted on the CPU, so for very high performance systems they are a good choice. Many systems of lower performance do not need them but may benefit. They operate by mounting a heat transfer chamber on the CPU chip and circulating a liquid (using a pump) through that chamber to absorb heat, out to a radiator mounted on an exterior opening, and returning it to the chamber. At the rad there are FANS that blow air to cool the liquid passing through. The most common arrangement is to mount the rad and and fans at a front intake opening with the fans sucking outside room air though the rad fins and into the case. Then there are more fans mounted elsewhere (top and back) to blow warn air outside. In the meantime the incoming air that has been slightly heated at the rad is also used to pick up heat from everything else inside the case (just by blowing around them) before it is exhausted by the fans. It is important to note that the heating at the RAD is typically a very few degrees, so that the impact on the air's ability to absorb and remove heat from all the other internal component is affected only to a VERY small extent, and is managed effectively merely by allowing the exhaust fans to alter their speeds. The speed of the INTAKE fans on the front-mounted rad is controlled entirely by a measured temperature inside the CPU chip so that the CPU's cooling requirement is "in charge of" the rad fans.