Should I peel turmeric?
To address a few points:


palatability, taste


There's not much taste to the "skin."  As with ginger, it's more of a texture thing.  For larger/more mature pieces, the skin can be thicker and perhaps create more of a texture contrast.  I think most people tend to peel it, but it's not strictly necessary.


food safety or hygiene


No significant food safety issue.  Hygiene is about the same as any root/rhizome vegetable.  That is to say: anything in the dirt will likely be on the skin, so if you don't want to eat the dirt (and what may be in it), scrub well or peel.  If you are concerned about possible pesticides or something else in the dirt, peeling is likely efficient for removing any on the exterior.

As for nutrition, after a few searches, I haven't found good information that implies any significantly different nutrients in the skin.