In my sentence 溶岩の中 / 上に、燃やさずにドアが置けるとは知りませんでした。, the subjects of 燃やさず and 置ける are both "I", and が is the object marker of the potential verb 置ける, as already pointed out. Therefore 燃やさずにドアを置ける is also correct. (And 置けるのは知りませんでした is correct as well. This は works as a contrastive marker, so 置けるのは has a nuance of "I already know something but didn't know 置ける".)
As for 燃えずに置ける, although this is a bit ungrammatical, I, either, don't think the subject of 燃えず is "I". Probably that's because 燃える usually isn't used for people. 焼かれずに would be used instead.
When the verb that is attached to ずに is non-volitional, the two subjects can be different in the structure ~ずに…する.
e.g.
最近雨が降らずに晴ればかり続いている。
不思議なことに、花も咲かずに実がなった。
~なくて or ~ず can be used for volitional verbs, but this is for explaining the cause/reason of the main clause.
e.g.
近頃うちの猫が餌をあまり食べなくて / 食べず、心配だ。(= 食べないので心配だ。)