No, apple juice does not contain dangerous levels of arsenic.
The US Food and Drug Administration (FDA) has been monitoring both the arsenic levels in apple juice, and the claims by Dr. Oz.
They have suggested it would be "irresponsible and misleading to suggest that apple juice contains unsafe amounts of arsenic based on tests for total arsenic". This is exactly what Dr. Oz has done, despite being warned earlier in a letter from the FDA.
[Source: FDA Letter to Dr Oz]
They explain that they have:
every confidence in the safety of apple juice.
and that there is:
no evidence of any public health risk from drinking these juices [...] FDA has been testing them for years.
Source: FDA Consumer Update: FDA: Apple Juice is Safe To Drink
The FDA resources go onto explain that organic arsenic has a far different toxicity to inorganic arsenic, and that measuring, as Dr Oz does, for total arsenic is misleading. It also explains the reasons for the difference between acceptable levels in water and apple juice: basically, water (including added to food) contributes to a far greater percentage of daily intake.
Is the FDA right about that? Well, the Agency for Toxic Substances and Disease Registry (ATSDR, part of the US's CDC) agrees that inorganic arsenic is toxic.
Source: ATSDR's Public Heath Statement for Arsenic
Inorganic arsenic has been recognized as a human poison since ancient times, and large oral doses (above 60,000 ppb in water which is 10,000 times higher than 80% of U.S. drinking water arsenic levels) can result in death. If you swallow lower levels of inorganic arsenic (ranging from about 300 to 30,000 ppb in water; 100–10,000 times higher than most U.S. drinking water levels), you may experience irritation of your stomach and intestines, with symptoms such as stomachache, nausea, vomiting, and diarrhea.
(They goes on to describe much more nastiness involved with ingestion and inhalation of inorganic arsenic.)
When it comes to organic arsenic, they are more vague:
Almost no information is available on the effects of organic arsenic compounds in humans. Studies in animals show that most simple organic arsenic compounds (such as methyl and dimethyl compounds) are less toxic than the inorganic forms. In animals, ingestion of methyl compounds can result in diarrhea, and lifetime exposure can damage the kidneys. Lifetime exposure to dimethyl compounds can damage the urinary bladder and the kidneys.
They contrast the length of time the two types of arsenic stay in the body:
Both inorganic and organic forms leave your body in your urine. Most of the inorganic arsenic will be gone within several days, although some will remain in your body for several months or even longer. If you are exposed to organic arsenic, most of it will leave your body within several days.