Can feedback loops occur in machine learning that cause the model to become less precise?
Yes, this is a real problem that manifests once system is used by real users.  

Most prominent example is News Echo Chamber (accentuated by ML based recommendation systems) 

ML algo sees that you like news / videos related to certain point of view, you watch more of such videos and model becomes more convinced of your choice. So it suggests even more content with similar views. 

https://en.wikipedia.org/wiki/Echo_chamber_(media)

http://theconversation.com/explainer-how-facebook-has-become-the-worlds-largest-echo-chamber-91024

https://www.theguardian.com/science/blog/2017/dec/04/echo-chambers-are-dangerous-we-must-try-to-break-free-of-our-online-bubbles

https://www.quora.com/Would-you-say-that-Quoras-generated-news-feed-suffers-from-an-echo-chamber-dilemma