What happens if one actually does go about eating the beads in those "do-not-eat" packets?
Usually, nothing happens if you eat silica gel. In fact, you eat it all the time. Silica is added to improve flow in powdered foods. It occurs naturally in water, where it may help confer resistance against developing senility. Silica is just another name for silicon dioxide, the main component of sand. Mayby you think if silica is harmless to eat, why do the packets carry the warning? The answer is that some silica contains toxic additives. For example, silica gel beads may contain toxic and potentially carcinogenic cobalt(II) chloride, which is added as a moisture indicator. You can recognize silica containing cobalt chloride because it will be colored blue (dry) or pink (hydrated). Another common moisture indicator is methyl violet, which is orange (dry) or green (hydrated). Methyl violet is a mutagen and mitotic poison. While you can expect most silica you encounter will be non-toxic, ingestion of a colored product warrants a call to Poison Control.
I don't think it would harm you: apparently silica gel is a widely-used food additive. According to Wikipedia:
Silica gel, also referred to as silica aerogel or hydrated silica, has FDA GRAS (generally recognized as safe) status, meaning it can be added to food products without needing approval. Silica is allowed to be added to food in the US at up to 2% as permitted under 21 CFR 172.480. In the EU it can be in up to 5% concentrations.
Listed uses include: anticaking agent, defoaming agent, stabilizer, adsorbent, carrier, conditioning agent, chillproofing agent, filter aid, emulsifying agent, viscosity control agent, and anti-settling agent.
However, obviously, you shouldn't plan on eating lots of it.
I did research on whether or not silica gel was poisonous, because I wanted to use a bunch of the silica gel beads as an abrasive method to clean out a bottle that was dirty. I added a bunch of silica gel beads and a bit of water into the dirty bottle to be cleaned, I put the lid back on and tilted the bottle back and forth and used the beads as a way of scraping the inside of the bottle clean. This did work well, but I found that upon adding water to the silica beads, quite a few of the silica gel beads cracked, this was from the silica gel beads absorbing water fast. The silica gel beads are designed to absorb moisture slowly from the air. The result of the silica gel beads breaking, were tiny slivers of sharp silica, so even if silica gel itself is not poisonous, eating them could be dangerous, because some of the silica gel beads will shatter into sharp pieces, and could cause cuts in the digestive tract, so avoid eating silica gel beads.