The attribute table field visibility settings are not saved if the layer's pop-up configuration does not contain attribute information.
Last Published: February 2, 2016ArcGIS Online
Bug ID Number
January 27, 2016
March 30, 2022
After review by the development team, it has been determined that this issue is related to a known limitation with the software that lies outside of Esri's control. The issue's Additional Information section may contain further explanation.
This is by design. To show or hide a particular field, change the pop-up, and the changes are saved in the web map.
1) Change pop-up contents to custom attribute display and enter in the HTML source view (simply adding a space in the default view reverts it to list of field attributes after saving).
2) Do not use a pop-up title and instead put the title in the custom attribute display in boldface.