XPS 8940, no boot, error Alert! Chipset heat sink not detected

Have you changed the cooler at all? I would open the case and inspect the cpu cooler/heatsink and see if anything is amiss.

This aluminum passive heatsink is the chipset heatsink.  Is it still securely fastened to motherboard?

And if it still seems correctly/firmly attached, try clearing BIOS:

1. Power off, unplug
2. Press/hold power button for ~15 sec
3. Open case and remove motherboard battery (check Service Manual for details)
4. Press/hold power button for ~30 sec
5. Reinstall battery (Time for fresh CR2032 3-volt coin cell battery? ~$2)
6. Close up and connect mouse, monitor and keyboard
7. Reboot

See if it's still attached.

On an old Pentium4-based Dell ... I had one of these get so hot that one of the loops in the motherboard heated-up and un-soldered itself . The heatsink fell off.

I had to remove the motherboard to re-solder it properly, but it worked fine after that.