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.