I don't know about all dumbphones, some models from last century (many of which won't function on today's networks because the radio standards they need are no longer deployed) aren't even smart enough to manage over-the-air firmware updates and it may be impossible to meaningfully reprogram those. But, like I said, if it doesn't speak a modern radio protocol it can't use the network so who cares?

I believe a Nokia 6310 is just barely new enough that it should work everywhere in the world, although I might be mistaken.

This phone (and many from that era) implement Java MIDP, a thin Java API to do some simple things commonly across different models of mobile phone. In particular we need to discover the UTC time (or at least a local time and then use a stored offset) and remember a secret, and then calculate the TOTP code and display it.

Getting a program onto these phones isn't necessarily easy though. It's not like they're running a web browser and you can just type in a URL somehow. You need to either have a dedicated program and a cable to connect it to a PC, or else you might be able to use WAP, a long obsolete simplified hypermedia protocol for these phones.