If you have access to a Project Collection in TFS 2015 but still not able to queue builds, then you are required to be added to the Project Collection Build Administrators group.

The error message would be as below:

500: Access Denied: <username> needs Use permissions for queue <queuename> to perform this action.

To give access, when you’re in the Source Control Explorer in Visual Studio and connected to the required collection, click on the Team -> Team Project Collection settings -> Group Membership.

Under the Security Tab, add the User or group under the Project Collection Build Administrators group as shown below: