Hi!

I've got my Red Prince character stuck slowed and burning.

Battle is long over, no gear could also ignite him, but he burns, without the digit "how many turns remain to burn", and restoration and help skills can't help -- he stops burning, but then ignites again.

Well, yes, he does. Is it possible that slowing came from one gear item and the burning from another?

Yeah, the only way to remove those effects is to wear the entire set at once or none of it. The entire set being cuirass, greaves, gauntlets, boots and helm.