A regular imprinted Tshirt uses ink.for the image/print. You can safely put a heat press on top of it. Just cover the image with a teflon sheet just to be safe that it isn't vinyl. It is ok to heat press vinyl too, you just need to cover it. The teflon sheets are cheap and you can buy them on Amazon. I just bought a couple a month or so ago, I think I got 3 large sheets for under $10. Are you using the wet pressing cloth to help adhere the fusible backing? If so, I would lay the wet pressing cloth down, tshirt on top, cover with teflon sheet and heat press away. The moisture from the cloth won't hurt anything. It will produce steam but steam is produced when I lay a screen printed shirt on it if I don't use the heat press dryer first.
I used to have an embroidery/screen printing business. I don't embroider for money now and don't have the large machines but I do still screen print often and have screen printing equipment. I use my heat press almost weekly.