I tried to trigger a powered minecart with dispenser and hopper, but no luck.
How can I give coal to it with a push of a button?
Minecarts with furnaces do not have an actual inventory, powering them is limited to "using" the cart while holding coal. It seems that the general rule is that anything that is operated like that (buttons, levers, picture frames, etc) is unaffected by redstone. Between this and the fact that neither the wiki nor any other source on minecarts with furnaces mentions powering them with redstone, it does not seem like there is currently a way to start them using redstone.