No effect if you are using wifi Internet instead of mobile data!
In fact, you can use TextApp without any need for a real phone number.
TextApp is a separate setup that requires only the Internet. So, if you are using mobile data, the use of the TextApp will consume some data, though the app is optimized. This may be reflected under description like ‘mobile data usage’ on your phone bill. The TextApp does not initiate any call or message through a real phone number, so no possibility of being billed additionally on a phone bill.