What are some ways to determine the cause of discs not spinning?

I have the original PS3 launch model (CECHA01) and it appears my Blu-ray drive has failed; it no longer spins any type of discs. But I'm unsure if this is due to the laser or a mechanical issue.

What's the best way to determine the issue if I'm looking to fix it myself?

Issues like this are covered by warranty, but if your warranty has expired, you can buy a bricked PS3 from eBay, or the like, and swap out the Blu-Ray drives. Whether that is a more feasible alternative to buying a new PS3 depends on how far away the bricked PS3 you buy is; over seas shipping tends to make it a very costly purchase, especially when buying things as large as a PS3 console.

The procedure of swapping the Blu-Ray drives isn't an awfully complex task, and you can find many, many references and instructional videos showing the whole process step-by-step on the web.

To appease to your actual question at hand: like Ekonion said, it's most likely a mechanical issue. Had the laser been broken, the plate would've spun up, and the PS3 would've reported the disc as unreadable or of unknown format.