A data refresh definitely shouldn't be causing a Tableau Public viz to break. Do you have a link to the visualization(s) in question? I might be able to look at the records on the backend.
I've had a similar problem. I was wondering if a Public 'Share' link is valid indefinitely or if it gets replaced over time. I know the main URL remains constant, but I'm not sure about the Share link.
We've added a load of links to a report and we've just noticed that a few of them are no longer valid. Here's a link to the report.
I've had to go through and double check all the links, so they work fine as of now, but if they are likely to expire, I need to put a disclaimer into the report.
Any help would be appreciated.