Hi Jason,
> Is there a way to view PDF feature attachments in a Portal Operations Dashboard, other than downloading it?
Displaying a PDF feature attachment in Dashboard is currently not possible, but this is a good enhancement request. Please submit an enhancement on the ArcGIS Ideas site for Operations Dashboard.
> ... is there a way to view it in a web browser
You could try and use the Embedded Content element, which allows you to display web pages in a Dashboard.
Hope this helps,