These two sentences mean different things and the difference is not a very small one at least in a place like this, if not anywhere else.
In the first sentence, 「何か」 is used as an adverb meaning "somehow", "for some reason", etc. You could not attach a particle to it even if you wanted to.
"There is somehow a burning smell." or
"For some reason, there is a burning smell (around here)."
In the second, 「何か」 is used as a pronoun meaning "something". To a noun or pronoun, you can, of course attach a particle.
"I smell something burning."
One more thing -- the pronunciation.
「何か」 in the first sentence, as an adverb that is, would often be pronounced 「なんか」 whereas 「何か」 in the second sentence would generally be pronounced 「なにか」 even in informal speech at least by the more careful speakers.
I warn you, however, that the author of the "best" answer in the second Q&A above does not appear to be familiar with the adverbial usage of "question word + か" words.