With external monitor: closed or opened lid?

when you leave the lid open, you can use the built-in keyboard and trackpad and camera, as well as using the second display for additional screen space in "Extended desktop" mode. Even if you use an external keyboard for typing, some users continue to use the built-in keyboard as a "Button box" for keyboard combinations with Imaging software

The MacBook Pro is primarily cooled by forced air moved in the sides and out the rear by the force of fans, whose speeds are adjusted based on measured temperatures. Additional heat build-up should result in additional fan speed, so accumulation of heat should not be an issue.

The amount of extra work to run a second display will not slow down any other operations, and almost that work is done in Hardware not being used for anything else.

A second display that uses DisplayPort (or its cousins ThunderBolt or USB-C display) may run cooler than one that uses only HDMI.

You can use your computer with an external monitor either with the lid open or closed. There is no noticeable difference wrt your concerns.

Use your Mac notebook computer in closed-display mode with an external display - Apple Support

Use external monitors with your Mac - Apple Support