I need to increase the Custom App Limit which is currently 10 in my Developer Org. The documents says that Salesforce does not provide assistance to the Developer Org. Can anybody please suggest some resolution regarding the same?
Unfortunately, it can't be increased in a developer org beyond 10. You can read here about the limits for various editions.
You need to have a Professional Edition or an Enterprise edition - in that case you can raise a support case with Salesforce to get this limit increased.