Latest iOS is 6.1 for the iPhone.
App no longer supporting iPhone 3G -wtf?

Not for the iPhone 3G - the last iOS version it supports is 4.21.
I've no idea why the app would require upgrading in order to continue running. It has worked perfectly well until now.
Not happy...

Isn't the main reason for making software updates mandatory to encourage people to buy new hardware?

Well yes - Apple do like to abandon support for devices that aren't brand new. My phone is 3 years old, so they don't care about.
But my beef is with chess.com - it's their update to their app that has cut me out. I'd love to hear what new feature they have added that couldn't possibly run on a phone that ran it before.

But my beef is with chess.com - it's their update to their app that has cut me out. I'd love to hear what new feature they have added that couldn't possibly run on a phone that ran it before.
Well, it is a free app so it is not like you lost functionality that you paid for.
A lot of times developers make the decision to not make the app backwards compatible because it takes resources to keep testing and supporting backwards compatibility.
The other reason is that sometimes a piece they used to make it compatible with 4.21 is no longer available in the SDK (although this is rare). Or, perhaps there is something in the SDK that takes care of a problem (like your battery drain issue) that makes it incompatible with 4.21.
I think you will start finding more apps that this affects as Apple wants to see the older phones go away. I think your beef needs to be more with Apple.

Maybe phones should have a "use by" date after which they just stop working (apart from emergency calls). Then at least one could plan ahead.

No, the iPhone 3 is still a good phone. But if Apple doesn't EOL them, they can't keep their sales numbers up.
I know several people that still have their 3 because it still works great.
So I've been using the app on my iPhone 3G for the last few months - it's a bit flakey and drains the battery but I've learnt to live with it as I enjoy playing live.
I just went to log on this morning and the App is telling me that I need to upgrade in order to play live. I go to the App Store and hit upgrade and am told "This app requires iOS 4.3". I have the absolute latest version of iOS installed on my phone that the 3G supports (4.21), so this upgrade is not possible. So basically Im out of luck.
Thanks a lot chess.com - I now need to buy a new phone to carry on using something that worked perfectly well a couple of weeks ago.