What are the consequences of dishwashering a cast iron skillet?

A cast iron skillet which is regularly washed in the dishwasher will be progressively stripped down to bare metal, which will quickly and consistently rust. It will not be usable for cooking until you clean it of the rust and re-season it.

Incidentally, I think you might be confused by the term "seasoning". Seasoning consists of polymerized oil, not "bits of food", and its purpose is to protect the skillet and reduce sticking, not to produce better tasting food. If you don't want to use a seasoned pan, for whatever reason, you should not use cast iron.

First, about the literal question in the title. I would go farther than Sneftel: almost no cast iron skillet is so perfectly seasoned that the metal is perfectly sealed. The seasoning is good enough for protection against air, but dishwasher powder dissolved in warm water is super corrosive to iron. Already in the first wash, you will likely get some rust developing underneath the seasoning.

Second, I am not entirely sure, but it seems that you believe that 1) a dishwasher will strip the seasoning, and 2) either a cast iron pan or a steel pan cannot be used at high temperatures or be put in the oven. Both these ideas are very much wrong. If you want to strip the seasoning, that is done with either lye, mechanical action, or in an oven's self-cleaning cycle. The dishwasher won't do it, unless you wash it frequently enough that enough rust builds up for the seasoning to flake, which is not what people want to get when they strip a seasoning. And you can absolutely put both a cast iron pan and a stainless steel in the oven or heat it to temperatures beyond what is needed for any kind of cooking or baking, without any damage to the seasoning of the cast iron pan.

Third, if you do have a cast iron pan without seasoning (which you seem to hope to achieve through the dishwasher, but you can't, see above), it will not only be stickier than a seasoned pan ("spray some olive oil spray" won't be sufficient, you will need a decent layer of oil), but it will also rust in your cabinet within a few days to weeks. Also, depending on how you use it, it will likely build seasoning on its own just by shallow-frying certain foods in it.

Based on experience with a carbon steel pan, which, as far as I know, is equivalent to cast iron in these regards:

If you wash it in the dishwasher regularly, and do not season it, you will just have a bare piece of non-stainless steel. This will have no non-sticking properties and will stick badly even if you add plenty of oil/butter for cooking; and it will rust quickly. At this point, you might simply not be using such a pan at all and switch to stainless or teflon.

The rust itself is no health concern. It does enter the food, potentially visible, and potentially with a slightly metallic taste.

The sticking is just... very bad. It would make me not want to use the pan at all. Note that you never leave the pan with "bits of food" sticking to it, that's not what seasoning is about at all. If your pan has stuff stuck to it, the way to clean it is to heat it up, and deglaze it with hot water (soap not required). Scrub with your metal spatula vigorously while the water is boiling, if needed. After getting rid of the water, let it heat up further until all water residues have vaporized. Then add a very thin layer of your oil of choice (canola, sunflower...), and keep heating until that starts smoking (and keep it smoking a bit). Wipe off any oil residues, and hang the pan to cool off.

This procedure cleans up any bits of food, builds up the seasoning a bit, and sterilizes the pan; it takes a very short time and should be a part of the regular cooking regime whenever it starts sticking.