Does leaving queries open in SSMS lockup memory on the SQL Server?
The only significant memory that would be held would be if SSMS was being run on the server itself (and then you'd still be limited to the 2GB of memory space that SSMS can allocate).

There will be a very small amount of memory allocated to maintaining the connection in SQL, but that's truly trivial.

Of course, if you leave an open transaction, that's a whole other level of pain, but it doesn't sound as if that's the trouble your developers are trying to avoid (unless this is what they are trying to avoid, but came up with a PC way of telling them that).