I believe the author of the plugin responded to a bug ticket 5 years ago:
"Seems it's a problem with access to nnn.dbf file, while it's already open by Qgis. As Qgis can't delete fields by the OGR Driver, Tablemanager has to modify it on his own.
Unfortunately I have no idea how to do it easily in operating systems not supporting multiple access to file (and I have no plans to invest more time into this plugin, as it's been designed as a temporary solution). I disabled this button on windows. I know it's not a real solution, so any help is wellcome."
Unfortunately, the issue still occurs in the latest v0.6.2.
I noticed that I also receive a second message immediately after saying:
"Saving successful. The old table has been backed up as example.dbf~. Do you wish to keep the layer style?"
If you leave the message and go to your TEMP directory (for me it's
C:/Users/Me/AppData/Local/Temp/), you will see the backup shapefile which has been correctly edited. Move this out of the TEMP folder before you click Yes or No in the message otherwise the shapefile will get deleted.
Ofcourse, this is a lot more hassle than just using the Save As... option.