I found a series of blog posts about the language of Barsoom here. It’s obviously quite speculative, because Edgar Rice Burroughs never tried to invent a consistent language. There is, apparently, only one complete sentence of Barsoomian in the series, and it’s one word long (‘Sak!’, meaning ‘Jump!’). But it’s interesting anyway.
For the John Carter movie, Disney hired linguist Paul Frommer to invent a more detailed Barsoomian language. He did the same for the Na’vi in Avatar. However the details of this version of Barsoomian aren’t available anywhere: it’s just something they used for the film. There’s an interview with him about it here.
Finally, there are some more speculations here, based on the premise that the language in Lin Carter’s Martian novels is the same as Barsoomian.