Loud fans during built-in Chess game, new Macbook Air 2020

Resetting the SMC requires a simple press and hold of four keys, you won't break anything. Macbook Airs 2018 and later have the T2 chip.

Resetting the system management controller (SMC) can resolve certain issues related to power, battery, fans, and other features.

Reset the SMC on computers that have the T2 chip

If your Mac has the Apple T2 Security Chip, follow these steps. If you need help, contact Apple Support. 

Notebook computers with the T2 chip

Before resetting the SMC, try these steps:

1. Shut down your Mac.
2. Press and hold the power button for 10 seconds, then release the button.
3. Wait a few seconds, then press the power button to turn on your Mac.

If the issue persists, follow these steps to reset the SMC:

1. Shut down your Mac.
2. On your built-in keyboard, press and hold all of the following keys. Your Mac might turn on.

• • Control  on the left side of your keyboard
  • Option (Alt)  on the left side of your keyboard
  • Shift  on the right side of your keyboard

1. Keep holding all three keys for 7 seconds, then press and hold the power button as well. If your Mac is on, it will turn off as you hold the keys.
2. 
3. 
   
4. Keep holding all four keys for another 7 seconds, then release them.
5. Wait a few seconds, then press the power button to turn on your Mac

The Chess app is indeed quite resource-intensive. It surprised me too. Open several games at once. Activity Monitor will show each game using 100% of each CPU core until it becomes task-saturated.

Activity Monitor User Guide for Mac - Apple Support

If that's your only concern there is no reason to believe anything is wrong.

The Chess app is surprisingly CPU-intensive. In fact, it's a great tool to use for stress-testing a Mac (make four copies of it and have the computer play against itself in each game and how well the Mac handles it before locking up will be a good indicator of how well your CPU/logic board is doing). The MacBook Air has an internal fan, but it's not connected to the CPU heatsink and therefore the CPU is technically being cooled passively. This has led to many thermal issues with the 2020 MacBook Air models, but especially with the i3 models. In fact, on MacRumors, there's a forum thread all about various modifications people have made to their MacBook Air's thermal system to better accommodate these thermal issues (doing this would definitely void AppleCare, but it's an interesting thread to check out from the standpoint of modifications that solve inherent design flaws).

My MBA with an i5 and 8Gb of ram runs the chess game with no problem. Did you reset the SMC?

Seems to be a lot of this with the new Macbook Air. Try resetting the SMC How to reset the SMC of your Mac - Apple Support