On a SQL Server 2005 instance in our DMZ, we have BIDS installed but not the SSIS service.
If I have an SSIS package on that server, can I still schedule to run it without SSIS service?
I tried using DTexec but it is not able to find the package.
The SSIS service doesn't actually do much and you can run packages without it or with it disabled. It's used by SSMS amd SQL Agent for example.
What you need are the base SQL Server binaries which are installed with the DB engine. So you need a SQL Server Instance of some flavour to have dtexec running. Whether it's the DB Engine or the SSIS service installed but disabled.
Running packages in BIDS is a special case. Run the package on the same PC as BIDS via dtexec and it will fail unless you have installed server components (= a SQL Server Instance)