Need new graphics card

First of all, you need to check what video cards your motherboard will work with.  Or you might need to update the video card firmware to work with more recent GPUs.   There's no point in going crazy with GPU specs if the rest of the system does not match it in perfprmance.   If you are notr sure what your current system specs are, apps like Speccy will tell you. 

I tried the brand new AI-powered Lens Blur in Lightroom, and my - that really put everything to work.

Granted, this was a 60MP file from an a7r V, and a fairly complex image at that, but still. This, I think, is the most machine resources used that I've ever seen:

Not only is VRAM maxed at 16 GB, it also starts using a couple of GB shared system memory. Never seen that before.

The math itself is mainly CPU here. Again, I don't think I have ever seen the CPU (i7-9700K in this case) maxed out this completely for this long.

Clearly, we have interesting times ahead.

"However, do the new AI features require or benefit from having more memory such as 8 Gb? Does the graphics card do the AI processing?"

I think you can test on your own. Run app of your choice, open Task Manager > Performance tab and monitor what is happening while working as usual. That will tell you whether you really need more GPU memory or not. If Dedicated GPU memory is filling quickly and often then you need more GPU RAM, if not...

If you get an RTX 4060 or 4070 you can't go wrong. They come with either 12 or 16 GB. That should be able to handle anything in Adobe apps for the foreseeable future.

Some of the new AI-based functions, like Denoise, run entirely in the GPU without touching the CPU. Adobe recommend a minimum of 8GB for Denoise, although I've never actually seen it use that much. It may be just a nicer way to say that you need a fairly recent model. The breakpoint seems to be Nvidia's shift from the GTX to the RTX designation.

For general processing, especially in Lightroom, VRAM usage may go up to 70-80% of whatever is available. I assume some sort of caching is going on there. I've often seen it stabilize at around 10 GB on a 16 GB card.