Reply by Tennessee

You can run glue through a planer, but I heavily recommend scraping off any blobs before putting it through. Dull them up very, very fast hitting those hardened glue blobs, and you might get some tearout if the glue carries some wood with it.
Titebond III is not that fast. What temperature is the room you are working in?
It is however very strong, and that is why you might get tearout on the planer in addition to dull blades if you do not take the blobs off before planing.

One side note: don’t forget, if you are planing with a blade planer, make sure all the grain is going with the blade rotation on that cutting board.

-- Paul, Tennessee, http://www.tsunamiguitars.com