Can it damage my iPhone 11 Pro battery if I use CarPlay while my iPhone is on a wireless charger?

If the vehicle has a wireless charger, does it have wireless connection to CarPlay? Check with the dealer. Otherwise, I'm not sure what the iPhone would do under those circumstances. I would think one of the charging methods will not activate. I'm going to go test mine to see if I plug it into the charger with the lightning port, what happens if I try laying it on my wireless charger and see what happens. I'm not using CarPlay, I'm just using the two chargers here in the house to see if there is any message on screen or other actions I can see.

I plugged my iPhone into the regular charger with lightning cable, it started to charge, showing a charging symbol, etc. I have a Belkin Qi wireless charger and laid the iPhone on the device, and the green light indicating it was charging lit up. I believe this would be a question best posed to Apple Support for an answer.

I don't know, but my gut suggests this is not a good thing to do at all. I'd recommend getting something like the WeatherTech Phone Cup Holder to put your phone in when you use Apple CarPlay.

https://www.amazon.com/WeatherTech-CupFone-Universal-Adjustable-Portable/dp/B07KY1XKRQ/ref=sxts_sxwds-bia-wc-p13n1_0?crid=3DDAD3TKN2HUD&cv_ct_cx=weathertech+cup+phone+holder+for+car&dchild=1&keywords=weathertech+cup+phone+holder+for+car&pd_rd_i=B07KY1XKRQ&pd_rd_r=b2f9383e-9930-49cb-ab33-7799b2538539&pd_rd_w=wBhvd&pd_rd_wg=SlQXM&pf_rd_p=5be4970c-0256-4afe-9550-68021bd84e5b&pf_rd_r=2PC88Q0A3KP9A516BTV6&psc=1&qid=1603817487&sprefix=weathertech+%2Caps%2C179&sr=1-1-791c2399-d602-4248-afbb-8a79de2d236f

You're welcome. Post back what you find so we can see and if anyone else searches and finds this question.