I'm working on a web application which is sending thousands of requests with application insights.
Where is Application Insights turned on and off from?
Where is exactly turned on I don't know, but I guess that you are looking for a way to feature toggle application insights, and to do so in your Application_Start you could do:
TelemetryConfiguration.Active.DisableTelemetry = true;
Doing this you'll stop sending telemetry info.
I hope this help, if it doesn't please let us know what you tried so I could have a better understanding of your question.
For more information you can check their documentation here. The section about Custom Initializers is also quite interesting and maybe will help you as well.