Are there benefits to learning to write with your non-dominant hand? There are some articles on the web that recommend learning to write with your non-dominant hand to get in touch with your inner child or a higher power, increase your creativity and be more open-minded or some such thing. I don't usually believe random stuff from the internet, but:

One of my drawing teachers told us to try drawing with the non-dominant hand. What I discovered was that I was very much better at getting proportions right with the non-dominant hand. I only had to use the dominant hand to clean up the shaky lines, because my non-dominant hand has poor fine motor control.

So there is obviously something that happens, when I use the "other" hand, something that has to do with how my perceptions are processed and my hand movement is controlled. But these "switches" while I draw are just temporary. What is sometimes recommended on some websites is a more fundamental change: to switch hands for all tasks.

Is there research on the effects of switching hands for all or certain tasks?

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Yes, there are benefits, but I don't think it requires long-term switches.  Studies have used this as a manipulation to try and increase self control and have found that it decrease aggression.  Based on this, once one has mastered using the non-dominant hand, it seems like the benefit of continuing to use that hand might be over (as it no longer requires effort).

See: Denson, T. F., Capper, M. M., Oaten, M., Friese, M., &amp; Schofield, T. P. (2011). Self-control training decreases aggression in response to provocation in aggressive individuals. Journal of Research in Personality, 45(2), 252–256. doi:10.1016/j.jrp.2011.02.001