I have described this problem and the process used to (unsuccessfully) fix it in detail at https://forums.zotero.org/discussion/comment/265112#Comment_265112. Basically, it seems the Zotero plug-in is successfully communicating between LO and Zotero, as 'Insert Bibliography' button works fine, but clicking on the 'Insert Citation' produces no result. This basically renders LO useless to me, as I rely on Zotero heavily. As detailed in the forum cited above, I have tried different versions of LO, Java, Zotero Plug-In, etc. I and a computer support person have done extensive web searches for a solution without result.
*** This bug has been marked as a duplicate of bug 104459 ***