Being in Natural Language Human-Robot Interaction (NLHRI), we do incremental processing. It increases speed by allowing processes to occur in parallel, and allows cool things like the robot looking at what you're talking about *while* you're saying it.
However, the biggest headache is that most NLP software isn't meant to be incremental, it's still mostly structured around the idea of reading in a whole document, processing the entire document, and then producing some results. Even algorithms that would be perfectly fine incrementally typically aren't programmed so they're easy to use that way. This results in a lot of needless duplication of effort when I finally get frustrated attempting to increment a non-incremental program and just write my own.