I just did a test on my phone to see whether that’s true (in my case), but it appears to be not I have wifi and mobile data both open and played a YouTube video. Before and after I checked how much mobile data was used; no difference. If you want to be 100 percent sure you can go to your settings and turn mobile data usage, per app, off. Depending on OS and brand, you most likely will find said settings under Wireless and Networks, then Mobile/Cellular Data, then Apps in Network (or something similar). In this tab you can choose WiFi and/or Mobile Data to be On or Off for all (pre-)installed apps. Hope that helps.