I set up a new AppService on Azure along with an empty database.
I downloaded and opened Orchard 1.10.1 in Visual Studio 2015. I right click the solution and rebuild. After that completes I right click Orchard.Web and publish. I downloaded the publish profile and loaded it into VS2015 during the publish process for the Orchard.Web.
The publish process completes successfully and opens a new browser window. After it compiles it only loads a white page with message:
Server Error in '/' Application. The resource cannot be found. Description: HTTP 404. The resource you are looking for (or one of its dependencies) could have been removed, had its name changed, or is temporarily unavailable. Please review the following URL and make sure that it is spelled correctly.
Requested URL: /

If there are no orchard logs in
site\wwwroot\App_Data\Logs, then you can check the event view logs to see if there is anything there which might be logged outside of the Orchard app:https://blogs.msdn.microsoft.com/benjaminperkins/2016/07/01/how-to-view-the-event-logs-of-your-azure-app-service/
Otherwise, rather than VS publish, you could try a console build and FTP?
Execute:
msbuild Orchard.proj /t:Precompiled /v:m /mThen upload the
\build\Precompiledfolder tosite\wwwrootIf this is a new site you should actually hit the setup page to paste in the connection string to initialize the empty database. Also this method should startup reasonably quickly as there will be no compilation done on the server.