It was quite of time ago the iconify icons introduced to dashboard 1.0 but I do remember there was a challenge to get them working and it took to have duplicate set of elements and then the switching was possible between those.
Direct replacing was not possible. I don't think that anything is changed since then. So I really doubt it is possible to have dynamic icons with Iconify icons.
Well as the "config-fields" document says - for button there is refresh required. So no support what so ever.
But what I tried to say is that Iconify is much trickier to handle and dynamics with them is hard or impossible to achieve with the dashboard widgets.
You may get some results if you try with the ui_template where you can figure out logic which is aligned with the iconify strategies.
I don't have a computer with DB1 installed nearby so can't even try out.
Well, I have a possible workaround in this situation as I use the button as a telltale (I don't need the button function, but it was an easy way to show an icon).
For some other stuff, I have use ui_svg which can dynamically change it's svg display.
Good point is that iconify icons are svg, so I can simply use the <svg> code from iconify website to inject in ui_svg node.
Still, it's a bit curious that we can dynamically change icons/colors of dashboard switches, but not button (even non iconify)