What happens to the effectiveness of the antenna if it gets covered in things like agricultural field dust, bird poop etc? Is it best to have this in an accessible location for regular cleaning?

There should be no major problem here.

We have not seen such issues in the past across many station deployments.  Poop and dust are transparent for RF.  We had an antenna with several inches of snow on it, and the signals were still good.  GNSS frequencies are in the 1 – 1.5GHz range which is a long way from ~2.4GHz where attenuation due to moisture is much bigger issue.