Here are few reasons and solutions that you can have a look why application insights aren't working after deployment.
SCENARIO - 1
Disabled Application insights on portal
Like @Vinez mentioned You can enable them from your appinsights overview page. Scroll down on the same page to see the left pane that shows an option of Application Insights and then “Turn On Application Insights”.


SCENARIO - 2
Missing AppInsights settings
You can navigate to Configuration -> Application settings and check if the appinsights settings (i.e.. APPINSIGHTS_INSTRUMENTATIONKEY
, APPLICATIONINSIGHTS_CONNECTION_STRING
, ApplicationInsightsAgent_EXTENSION_VERSION
, XDT_MicrosoftApplicationInsights_Mode
) have been added or not with appropriate values.


SCENARIO - 3
ApplicationInsights.Config not deployed with the application.
You can go to App Service Editor (preview) and check whether the AppInsights.config file is being deployed.


You can check this similar thread which deals with build and publish.
SCENARIO - 4
Due to restricted firewall.
Please try checking Enable Azure Application Insights on the VM-Series Firewall
Further you can log the traces and act accodingly using perfview.
REFERENCES :
- App Insights for web application not working when deployed to Dev server
- Troubleshooting no data - Application Insights for .NET.