If you are now using the latest Map Viewer (formerly Beta) and can not find the Refresh Interval setting (it is apparently coming), then this solution worked for me.
- Log into ArcGIS Online Assistant (https://ago-assistant.esri.com/)
- Under the menu 'I want to...' select 'View an Item's JSON'
- Click on the Map Viewer map that doesn't have the refresh interval options
- Scroll down to your Data area
- Click on the Pencil (Edit JSON option). You may get a warning the first time about undesirable results ... I would recommend you make a copy of your map first so that you do not accidently break something
- Add in the following line (quotes and all) to the JSON for each layer that you want to have refresh "refreshInterval": 1, (Note: I placed mine above the line that has "layerDefinition": and the number 1 is 1 minute)JSON edit
- Click the 'Save' button if when hovering over it, it tells you that your JSON is correct.