I finally made the effort to purchase the ATI Radeon HD 4870.
So far, it has totally eliminated the crashes and Maya runs quite smoothly, even though the card is not officially supported by Apple on the 1,1 Mac Pros.
Thanks again for the help everyone! I can finally do work for my internship without having to worry about Maya crashing!
Mac Pro 1,1 - Mac OS X 10.5.8
5 GB DDR2 RAM
2 x 2.67 GHz Dual-Core Intel Xeon
ATI Radeon HD 4870