Is it impossible to read and write NFC tags in my Galaxy S10 series while using LED cover case?

This means that as long as the LED functionality of the cover is used, you cannot use NFC feature. However you can use NFC again after LED lighting is off. Please refer to pop-up presented above appeared when Mode light or LED Icon is activated.

FYI: Galaxy S10 LED cover is a new type of official accessories. It is a back cover with some unique LED features.
 

•Mood lighting : When Galaxy S10 is facing downwards, a light show is presented
with the LEDs. 

•LED notification icons :  You will get
notification icons on the back when a notification comes.

•Camera timer : If the timer for the rear camera is turned on, the countdown
will be shown on both the screen and the cover.

•Emoticons for the rear cameras : When you open the camera app, and the rear
cameras are active, a LED icon you chose will be shown on the back cover.