Ingestion of hydrogen peroxide (H2O2), especially very high strength H2O2, can be very dangerous and can cause some serious health risks and possibly even death.
A United States Food and Drug Administration (FDA) announcement made in 20061 says that drinking high strength H2O2, specifically H2O2 of 35%, is extremely dangerous and can cause several serious side effects. "Ingesting hydrogen peroxide can cause gastrointestinal irritation or ulceration." It can also cause other health risks, some of which can be life threatening.
Another article from 20072 talks about a specific incident in which a 39 year old man accidentally drank 250mL of 35% H2O2. Though the man did not experience the worst possible side effects, he did have to go to the hospital and he did experience damage in his stomach. This is a good example of a real life situation in which H2O2 was consumed, and it did not turn out well. Another real world example3 turned out much sadder. A 2 year old girl swallowed two sips of 35% H2O2 and died. This happened because of a cytotoxic (cell-killing) injury in the tissues and formation of oxygen gas (oxygen toxicity) caused by the H2O2.
Ingestion of 35% H2O2 is undeniably dangerous, but what about lower concentrations of H2O2, such as 3%? A study from the 1990's4 studied 670 cases of mostly children being exposed, usually orally, to 3% H2O2. "Exposure to hydrogen peroxide 3% is usually benign, however, severe gastric injury may occur following small ingestions in children." Overall, they found that a majority of the children were not affected by the low concentration H2O2, but there were special cases in which bad outcomes did occur.
Overall, we can see that safeness of drinking H2O2 varies depending on what the concentration is. High concentrations, 35%, are extremely dangerous and should never be consumed, but low concentrations, 3%, present low risks. Now, the question is do the benefits of 3% H2O2 outweigh the risks.
I have been unable to find any reliable sources showing that health benefits of drinking H2O2 of any concentration. Usually, it is used as a topical solution for minor cuts and wounds, not as something to be taken orally. Though the ingestion of 3% H2O2 hasn't been shown to be consistently dangerous, there have been cases of damage in the stomach and other parts of the body.5 H2O2 poisoning can be very dangerous, even with low concentrations of it, so I would not recommend drinkin H2O2, ever. For more on Hydrogen Peroxide Poisoning, see here and here.
 FDA Warns Consumers Against Drinking High-Strength Hydrogen Peroxide for Medicinal Use: Ingestion Can Lead to Serious Health Risks and Death
 Accidental ingestion of 35% hydrogen peroxide
 Fatal accidental ingestion of 35 % hydrogen peroxide by a 2-year-old female: case report and literature review
 Hydrogen peroxide 3% exposures
 Hemorrhagic gastritis and gas emboli after ingesting 3% hydrogen peroxide
Hydrogen peroxide poisoning
Hydrogen peroxide poisoning