With no HDD you have no operating system, and no GPU drivers. A lot will depend on what options were set in the BIOS. If it can boot from DVD you can run a Linux Live DVD and run from there. Select try Linux instead of install and it should run w/o an HDD. Or install it to HDD if you like. There should be a COA sticker with an activation code for the original OS on the computer somewhere. You could then install and activate that version of Windows. if you get that far you can go to Dell and get drivers, and BIOS updates.
http://www.dell.com/support/home/us/en/04/product-support/product/precision-t3500/drivers
They have them for Win7,Vista, and XP.
To try and enter the BIOS toggle F2, to try and boot to Windows Safe Mode F6. Any GPU should run Safe Mode w/o drivers.
Her are some basics to check. Do you have a keyboard plugged in? Is there a GPU installed and connected to the monitor? There are 2x GPU slots try moving the card. Does it have 2 GPUs?
BTW congrats on the T3500s. They have good potential for a budget gaming rig.