I may be wrong, but I'm not sure Acrobat can do that. There doesn't seem to be a B/W colourspace (there are plenty of greyscale ones) that you could convert the colours to.
How many pages is it? You might be able to delete the background coloured "box" in Illustrator, but that would require doing each page as a separate file and recombining.
Another possibility, depending on the complexity of the layout, might be copying or exporting the text and making a new PDF document.
If you're on OS X, Apple's ColorSync Utility can apply a Black/White filter to a PDF.