The short answer is no, unless you are using a very unusual setup (such as using 3G or LTE cellular data modem for your home router internet connection.)
In a normal setup, your router is connected to the ISP internet, and your cell phone or tablet has to be switched to router WiFi in order to see and control the Lumin player. So your router WiFi and the ISP internet connection is used.
To be more safe, you can turn off iOS WiFi Assist (which may use your cellular data even if you think you are using WiFi), and/or explicitly disable Mobile Data for the Lumin app and all other music streaming apps (if you only use them to AirPlay to Lumin at home).