The problem, I think, is the nature of cursive writing. It's a relic of an earlier age, when never taking the pen off the paper was necessary to avoid unsightly ink splotches or smudges, and while it's true that it can be faster, it's that much harder for other people to read. The general impression I get today is that, for better or for worse, it's on the way out - not only is it not a priority for schools anymore, handwriting as a whole is being nibbled at by electronic alternatives. Five years ago, I made all my university course notes on paper, but even then there were people here and there tapping them into laptops instead - and that was before ubiquitous smartphones or iPads.
The implication, then, is clear: cursive writing is just the canary in the coal mine, and sooner or later the practice of writing in general will decline, as electronic and voice-recognition transcription systems make it possible for people to record their thoughts and ideas without ever having to resort to paper and pen. Clarity would improve - from millions of different hands, subtly shaped by the way each writer learned from the examples hanging over the blackboard and the way they held their pen, we would go to a relative handful of fonts designed for legibility. One prime reason that I never put an effort into learning cursive was that it would be counterproductive for me for exactly that reason. I've never come across anyone who could not read my printed handwriting, whereas if you go by my cursive chickenscratch, you would think my name was something like "Aw Batv."
On the other hand, the replacement of traditional writing methods with electronic ones wouldn't do much in terms of saving paper - if anything, it would encourage more use. There will still be a need for physical copies of documents, and running something off on a printer is a matter of pushing a few buttons; no exertion at all. Writing or no, the occasional dream of a "paperless society" is still a long way in the future, and until we develop methods of perfect information security and ubiquitous hardening of storage media against electromagnetic pulses, it's better for it to remain a dream.
Will writing become a lost art? I don't think so. There's a great many stages between ubiquity and irrelevance, and handwriting is just too valuable to lose completely. What might happen is that the frequent use of handwriting may end up becoming a signifier of affluence, effectively allowing a person to say "I am so well off I can afford for this task to take several times longer than would otherwise be the case." Though I don't think the idea of writing would become unheard of, I think it's likely that outside certain specific situations, people might come to think of handwriting as being a bit odd and old-fashioned.
Still, it's the most significant invention in the history of humanity. The future of writing will greatly influence the shape of years to come.