Salesforce permissions plays a vital role in using the Developer Console.

Please check the following for Salesforce permissions you need to have Developer Console work.

To use the Salesforce Developer Console:

API Enabled AND View All Data

To view, retain, and delete debug logs:

View All Data

To execute anonymous Apex:

Author Apex

To use code search and run SOQL or SOSL on the query tab:

API Enabled

To save changes to Apex classes and triggers:

Author Apex

To save changes to Visualforce pages and components: 

Customize Application

To save changes to Lightning resources:

Customize Application