By default, your AppServices run in UTC. There is an almost undocumented (or at least very little known) feature of KUDU which allows you forcing a specific time zone to your application. By adding specific Application Settings to your App Service (in Azure Portal) you can set following features:

• WEBSITE_TIME_ZONE, e.g. Central Europe Standard Time (List of supported values)
• SCM_TOUCH_WEBCONFIG_AFTER_DEPLOYMENT = 1/0 – does what it says (default is 1)
• SCM_TRACE_LEVEL = 1..4 – you can get more detailed tracing by setting higher level (default is 1)
• Build/GIT related options
• Caching related options
• Setting Node/NPN version
• WEBSITE_LOAD_CERTIFICATES – Loading certificates
• Low-level diagnostic switches
• WEBSITE_DISABLE_SCM_SEPARATION – allows you to run your site and SCM (Kudu) in the same sandbox (and potentially save some resources) – unfortunately marked as legacy-obsolete and not supported any more
• and many more

You can find complete list of features in Project KUDU Wiki.