The two important points:
1. The Mac Pro will work just fine with one processor removed.
2. The quad-core chips in the Mac Pro are Xeon 5300-series chips that use Intel's "LGA-771" socket. This is not at all compatible with the more desktop-oriented "LGA-775" socket used by Pentium 4, Pentium D, Core 2 Duo, and Core 2 Quad chips. The Xeons only work in Xeon sockets.
However, if you had, say, a first-day-of-purchase bottom-of-the-line quad 2.0GHz Mac Pro (two 2.0 GHz dual-core chips,) and your spanking new octo 3.0GHz Mac Pro (two 3.0 GHz quad-core chips,) you would see a performance boost in the old machine by moving one of the 3.0 GHz quad-core chips over. (You'd end up with two 3.0 GHz quad-core machines instead of one 2.0 GHz quad-core and one 3.0 GHz octo-core.) But you'd be "wasting" the two 2.0 GHz quad-core chips. It would only be even remotely useful if you did a lot of work that was capable of using all four cores, but not eight cores, on both machines. In which case, the speed loss to the octo-machine wouldn't be felt, and the speed boost to the old 2.0 GHz machine would be.