I am working on an existing map in Portal. I created the map weeks ago, I own it, and I am signed into my Portal. I made some changes today, and now I see there is no Save button on the map interface. I tried signing out and in again, I tried Chrome, IE and Firefox, all with the same result.
I am having exactly the same problem using IE11 (our corporate browser) and I share all the same circumstances you have. However, Chrome works for me. I have not tried Firefox.
Have you been able to fix the problem?
Make sure that you do not have both secure and unsecured content in your web map. Go to ArcGIS Online Assistant and log-in with your admin credentials to your portal. There you can easily make sure all your layers are either "HTTP" or "HTTPS". If there is a mix, you will get errors like the Save button not showing up in your web map.
I've seen this issue once with a user who was adding a KML Layer from Google Drive with the link being registered as an item in Portal.
In order to eliminate something similar to corruption, could we re-add each layer in the original webmap individually in a new webmap, save them and then go back out and re-open the map in Map Viewer to see at what point the Save button goes missing? I know this can be cumbersome if you have many layers, but I suspect that there could be one or two layers that have a missing reference in them.