In my case the culprit was a lightbox implementation of the JotForm feedback. It loads some kind of external javascript which in this case disabled functionality of the tabs and spoilers (I don’t know why it did this). I changed this function to a popup to disable the externally loaded javascript.
The Jotform feedback was something which I added manually into the theme files, so a simple check to see if it is your theme files that are problematic is to just put a random tab or spoiler function on your start page and live preview another theme. If the tabs work, then something in the theme you are using is causing the problem.