Short answer: Nothing extraordinary, if you use common sense.
Long answer: It is not recommended that you try to swallow it. (Sorry, small joke.)
Don't try to take apart either NiMH or LiIon batteries, because the contents are volatile and poisonous. Definitely don't try to charge either kind of battery by directly connecting them to a "dumb" current source (i.e. a simple power supply) without the phone being involved. This is particularly a problem with NiMH batteries, because they can overheat and become a fire or explosion hazard. Unlike NiCad or Lead Acid batteries, charging of LiIon and NiMH batteries is a science and is more complicated than simply back-flowing current.
Using a phone while you are driving can distract you from the road, which can increase your chance of a traffic accident. Particularly, dialing a phone or looking up a phone number in the phone book can be a problem, since that usually requires your eyes. Answering a call and talking is usually less of a problem since many phones permit "any key answer". But a car with a manual transmission requires both hands to drive safely, so even using a phone in such a car can be a hazard. A hands-free carkit will largely eliminate this problem.
That said, most people who ask this question are concerned with radiation, and by far the best source of information about that is this page: Cellular Phone Antennas and Human Health. (While I recommend that you read this page, what it comes down to is that there is no confirmable danger from cell phone RF. There is a lot of rumor and speculation and unconfirmed suggestions, but no confirmation.)
(3/2000) Innuendo and fear-mongering can make almost anything sound like a threat to health. I've found a beautiful web site which demonstrates this: the Dihydrogen Monoxide Home Page. Go read it, especially if you don't know the common name for DHMO.
There has been advertised a product which claims to decrease the radiation broadcast by a phone by covering it up. For my opinion on that, see this.
"Dihydrogen Monoxide" means a molecule consisting of two Hydrogen atoms and one Oxygen atom; in other words, H20, more popularly known as water. What's amazing about that page is that most of the dangers they describe are real. Many people do die from water inhalation each year, for instance; we call that drowning. Equally, many people have been seriously injured by "gaseous DHMO", more popularly known as steam.