Yes and no.
The 100% safe answer is no. You’ll notice in the sidebar I suggest buying a dyeing-only salad spinner if you do this a lot and want to cut your drying time. With a spinner, the dyed fiber is in direct contact with the plastic and who knows what might soak in? Don’t use it for food again!
When I put the plastic-wrapped fiber packs inside the metal pot, though, chances are pretty low that dye is going to touch it unless one leaks… I have a dedicated pot I use for dyeing, but given a thorough scrub, I’d probably be willing to put food in it. That’s me, though — I have an iron stomach, no pun intended! If you try this once, I don’t think you’re going to contaminate your metal pot permanently.
However, since this article’s photography was done, I have made the move over to using Ziploc bags and plastic takeout containers. Why? For one, you can dye a lot more stuff at once! With careful sealing and positioning, you can dye 20+ yarn or fiber packets in one dishwasher cycle instead of just a few. The point of the pot + lid was to keep excess water off the fiber and build up extra heat around the packets. Having multiple packets in one load creates the same effect.
You can use different sizes of Ziplocs — from sandwich-sized tester dye packs to 2-gallon superjumbo storage bags dyeing a sweater’s worth! Also, our local Chinese takeout uses supersturdy plastic containers that are even stronger than Gladware. They’re good for both the dishwasher and test-dyeing in the microwave. I’ve even used Reynolds oven bags to dye in the oven!
In short: I’d recommend the Ziplocs if you’re not willing to devote a dedicated pot to dyeing, and I’d give everything a good bleach-filled scrubdown afterwards (as I do).