These were never shipped with the Core 2 Duo, so even if the board accepts it it may have weird issues. The Pentium D/4 issue is even gray until you know which board you have, as mine had the 0WG261 board. The board mentioned above is known to be Pentium D incompatible unlike the E510 board, which made it’s way into the later 5150 systems (even with the Pentium 4 only limitation removed!). Since it didn’t complain, it sounds like you have the good board from the E510 which does put you in a better position. The E510 board also made it’s way into the OptiPlex GX620 as well.
If you wanted to try it and see long term (but DO NOT rely on it yet), swap the power supply for a quality unit like Seasonic - the OEM Dell power supply is known to have capacitor issues.
I haven’t looked at it too closely, but I know after these machines Dell did release the OptiPlex 745 which DOES officially have Core 2 Duo/Quad support. It may very may well use the E510 board as a base, but it has a different chipset (Q965) instead of the i945. If you try and flash the 745 BIOS, you will kill the machine if it doesn’t get blocked anyway, so do not even try it. The chipset change does not shock me since Dell did finally add SATA II support in 2007.