An open query window in SSMS does take up some resources, particularly there is an amount of memory allocated for the connection. However, for a single or even ten connections, this is nothing the DBA should worry about. When this can be a problem is if an application leaves thousands of connections open.
You say that you fill up temp tables in your first session. These temp tables take up space in tempdb, and if they are several gigabytes in size, the DBA may be concerned. Whether he/she should be concerned, I don't want to make a judgement of with the little I know at this point.
What absolutely can be a problem is if you do:
BEGIN TRANSACTION
-- Do something
and never commit that transaction. You would be holding locks that could block other users. But there can be problem even if you would only operate on your private tables, since that open transaction log prevents the transaction log from being truncated.
There is actually an option in SSMS that can help you to make friends with the DBA. Tools->Options->Query Execution->SQL Server->Advanced->Disconnect after the query executes. However, if you disconnect, you would lose those temp tables, so it does not sound like it would suit you.
As SimpleSamples say, you need to work with the DBA. Maybe there is a real problem that the DBA has not been able to communicate clearly, or which did not register with you, because you may not have the technical understanding to SQL Server to grasp it. Or maybe the DBA is just over-zealous. All this is hard to tell from a distance.