G'day and welcome to the forums.
From your all too brief description, it sounds like you have a nVidia 7300GT graphics card installed. They were the weakest and cheapest when the Mac Pro was new, and have not improved with age, heat etc. They can do precisely what you say and do you have anyone you know who has a Mac Pro? the shot would be to remove your card, firstly give it a very good clean and the PCI-E slot, re-install it making sure it i in firmly, connect up and see if it will boot. If no joy suggest looking for an 8800GT series II card. The 8800 cme out for the Mac Pro3.1 2008 model, but this card was later released for the Mac Pro 1.1, and is a better card. Of course the problem is after lashing out your money, it may benever problem as the 1.1 model is now over eleven years old, approaching the dinosaur age.
When you post, pop in all relevant information, operating system, etc. I will search eBay and include a link to the correct 8800GT.
http://www.ebay.com/itm/USED-1st-Ge...00-GT-512MB-Video-Graphics-Card-/252214463328
That is the card. The cost is probably about the value of the Mac Pro alas, but what else can one do?