The real risk isn't really to the toaster (who cares, it's just a toaster, and even if it starts on fire when you plug it in, you're probably watching it pretty closely).
No, it's that you leach lead or other such nasties (sealing rubbers, PCB epoxies, what-have-you) into your dishwasher and start eating them when they get redeposited onto your dishes. Lead isn't that harmful... except when you eat it. Don't eat lead.
In fact, if you have a dedicated dishwasher for this (or ultrasonic cleaner! those are pretty awesome, most of the time, for most things!), I'd say it's perfectly reasonable to do this! You know what we call those kinds of things where I work? Parts washers. You know what we wash in them... circuit boards! It's fine! Circuit boards are cleaned with aqueous solutions all the time as part of normal manufacturing! If you don't have a dedicated dishwasher... well, I wouldn't do it then, but if you must, you'll want to run the dishwasher a few more times, empty, with a strong detergent, preferably a lead-chelating one. (You might also make one of those runs with dishwasher cleaner, and strike the whole "yearly dishwasher cleaning" task off your to-do list for the first time this... uh, decade?)
You will, of course, want to dry things well. This is both easier and harder with a toaster. Easier because, well, toasters get hot, and this will get any nooks and crannies dry. Harder because one of the tricks to get things dry faster after they come out of a parts washer is to rinse them in alcohol, because alcohol dries very fast. You, uh, don't want to do that with a toaster. At least not one you intend to run any time soon.