Having spent half an hour trying to get the Zotero integration in LibreOffice to work (getting LibreOffice to detect Java needed the libreoffice-java-common package by the way), the toolbar appeared and worked, but without icons.

Getting the icons to work required the following:
First, close LibreOffice.
Second, run the following commands:

rm -r ~/.config/libreoffice/4/user/temp/
rm -r ~/.config/libreoffice/4/user/extensions/

Now after starting LibreOffice, you should see the icons instead of just words for each button on the Zotero toolbar.