Do I have an IP address if I'm using the cellular data on my iPhone?

For example: If I were to visit a website which collects IP addresses, what would show up if anything at all?

In a 3G/4G network every device gets an IP address, but THAT's NOT the IP address that you see when going to sites like www.whatismyip.com. That's the address that the Telco presents to the external world, not the device IP address.

Telcos such AT&t, Verizon, Telefonica and similar assign a "private" IP address that is only valid in their network. This is similar to the internal IP address that you have in your phone when connect to the house wireless, but if you check in www.whatismyip.com you get the external IP address of your wireless router (You can check that those are different addresses). What Telcos do is known as NAT or PAT. The reason is that the current version of IP has a very limited number of available IP addresses, and all those million of devices can't get public IP addresses (like the one you see in whatismyip.com). Actually several devices share that external IP address.

Unlike Android devices where you can get the IP that the telco assigned to the device, iOS does not present that information to the user (unless you jailbreak the device or have an App).

Although the address that whatismyip presents is not your real IP, it is the one that the external world recognizes so it suffices for most purposes

Yes there's always an IP linked to your phone.