Can I cancel my U-verse internet package but keep the TV and still have it work?

What @skeeterintexas said is correct.

You don't need AT&T internet, but if you have HBO Max grandfathered in, you'll lose that.

You also need AT&T internet for the music channels.

Keep in mind that you could lose various promotional prices if you drop AT&T internet.

And yes, you'll need the AT&T gateway and WAP. The gateway connects your U-Verse TV to the outside world. The WAP connects your gateway to the wireless receivers.

The gateway connects your U-Verse TV to the outside world.

I would clarify this to say that the Gateway will connect your U-verse IPTV to the AT&T U-verse IPTV Network--but not the Internet once you disconnect Internet service.

Technically, U-verse IPTV is not Internet traffic.  When you have AT&T Internet and U-verse IPTV they share a physical connection through the Gateway.  When you discontinue AT&T Internet, then you may maintain the U-verse IPTV service, but you'll need to keep the AT&T Gateway and its connections to the WAP (if you have one) and the DVR and any other wired receivers.  

This means all this wiring is reserved for the AT&T U-verse service and cannot be used (at least without some significant work) for your Internet service.

Thank you