I don't think there is a down side to emailing, although if they are not communicating with you it is not a good sign for your prospects. Of my 5 faculty rejections, 3 emailed me or called me, and 2 required me to email the search chair.
A possible email could use the language below. Parts of the email that are optional/circumstance-specific are in brackets.
Good afternoon! Thank you again for having me out to visit SCHOOL in January. I am writing to inquire about your decisionmaking timeline. [OR I am writing to let you know that I have received an offer from another institution.]
[My timeline for accepting this other offer is approximately X. I wonder if I could receive a response regarding your search within that time frame.]
[Is there anything you can tell me about the status of the search? From the timeline that you mentioned during my visit, I surmise that the school has made an offer to another candidate. Has that candidate accepted, or is the search still ongoing?]
Thank you once again for considering me for the position. I look forward to hearing from you.
Thanks to Karen Kelsky's book The Professor is In for some of this language.