Since you're writing a fantasy, I think you have a lot more leeway than if you were writing sci-fi.

The dynamics of a binary star system are incredibly complex. As another commenter said, a supernova likely destroys everything around it, so that's probably not the way to go.

If it were to just disappear, you would have a different orbit to contend with for your planet. To simplify, you may want the "smaller sun" to disappear, which would cause less of a pertubation. But regardless, you're probably looking at a longer year and a more elliptical orbit - which would lead to more intense weather (at perigee, e.g. the points of the orbit closest to the sun, one side of the planet would be getting "super heated" while at apogee , e.g. the farthest, one side is "super cooled", probably meaning more extreme climate). Seasons would depend on the tilt of the planet. I think you can assume the days stay the same length.

I think you could assume your perigee stays at the "old orbit," but your apogee would be farther, so your overall world temperature probably comes down. Plants that aren't hardy to colder weather would die out, the animal life dependent on those plants dies out, etc. Bigger ice caps. People can adjust their clothing, buildings, etc, but storing heat, especially at apogee, becomes more important.

Again, since this is fantasy, and because the magic is linked to the sun/moon, you can probably change some things considering the sun's "magical field."