Chop it up and swap it around. See if the meaning is still there.
"Anything he said, I didn't pay attention to it." The point is, you didn't pay attention. I'd really have to stretch my imagination to accept B as a possibility.
The basic idea of the sentence is that you were ignoring him. A double-meaning doesn't jump out here.
The only alternative reading I can think of is: "I didn't pay attention to any thing he said." 'Any thing' meaning 'any particular thing'. So you paid attention, but not what he said. Maybe the tone of his voice, or that he needs a shave, or that the sky was very blue that day... so you got some kind of impression, but little to do with his words.