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In Futurama every person has a chip implanted with their personal details and a predetermined job.
On the show the chip seems to be implanted in to one of their hands, and is easily removable.
Where would be the best place to have such a chip installed at birth that would be extremely difficult to remove?
Inside the skull
When babies are, well, babies, the skull is actually rather soft. Right in the inside of the skull would make a chip very hard to remove, but not incredibly difficult to insert.
Near an Important Artery
As a baby, arteries are much smaller and much more accessible to surgeons. A tiny incision could take a chip right close to major arteries with little risk. In the future the tiny chip remains tiny, while these major arteries grow considerably. It would take a skilled surgeon to remove the chip without damaging these vital arteries.
The authority responsible for placing the chip could also vary the locations of these chips to hit various arteries, so finding its location would factor into the cost of having it removed.
Within Bone
Putting the chip inside of a subjects bones could also do the trick. Removing it would be incredibly painful (think bone marrow transplant) and costly. There are also many bones in the body, so this also provides an opportunity for varying the chip location.
The Real Issue
Regardless, if the chips can be inserted by a skilled surgeon, there must be the possibility of someone with similar expertise removing it. The real deterrent would be laws against its removal with regular checks (and likely strict punishment), and difficulty of removal. Making the actual procedure for removal incredibly difficult and expensive allows only the most wealthy or skilled to actually remove the chip.
Rabbit's suggestion of inside the skull is good, another possibility would be in the chest cavity. It could be inserted with a thick needle and the surgery to remove it is actually more difficult than opening the skull (if you don't get into the brain itself). You have to cut through muscle and bone and avoid a bunch of blood vessels that you really don't want to mess with if you don't know what you are doing.
Make them breath it in
If your chip is going to be around similar dimensions to that in Futurama you could make people just breath it in. It will end up somewhere inside the lungs where it'll be nigh impossible to remove without extensive surgery.
The added bonus is that this way allows to even chip people that were born off the grid or outside your sphere of influence.
So you want the chip to be hard to remove. Right. Does it matter if the subject survives the retrieval?
Get me that chip, I don't care. Put it somewhere accessible, but hardish to get to if the subject does not want you to. Go for between the shoulder blades if you want the least hassle. Somewhere under the skin in the breast area for visibility. Maybe in the buttocks? (not many nerves there)
I don't want to hurt or endanger you. And seeing you got your chip implanted into one of the upper vertebra, as in, it's part of the bone, it will be very hard to remove without complications.
(See Logans answer for some ideas. Or in skull like Rabbit says.)
Here's a radical concept; nanobots.
These nanobots would be injected into the blood stream and use genetic information as a biological identifier and broadcast relevant information to the parties with access to the broadcast codes.
There would only be a few nanobots and they wouldn't need to store very much information. In fact, the data could be spread across a few bots if data storage is a problem. The reason why this is an excellent option is because there would be no way of knowing where in the blood stream the nanobots are hanging out.
While this isn't a chip per se, it serves the same purpose and is basically a chip that was chopped into a bunch of tiny pieces and still works.
I think something that could be quite important to think about here is also that you don't want the populace to lose the chips implanted inside of them. There are quite a few job or hobby related incidents in which people lose digits are even entire limbs, like their legs or arms. Because of that, it could theoretically be considered a bad choice to place the chips here, unless of course you go with the idea that the government predetermines the chipped people's jobs as well.
In that case, you could decide that people in low risk jobs could get implanted in their arms or legs, and other people could get implanted somewhere else. It's that "somewhere else" bit that gets even more opinionated. Like others have stated, while it would be arguably easy to place a chip inside the head of a baby, it wouldn't necessarily be easy to pull the chips out later in life. A chip placed in front of, or near, a vital organ could result in pieces of the chip doing damage to the organs themselves if the chipped person is ever shot or stabbed in that area. Likewise, putting the chip in other areas, like under the arms or lower belly area could make sitting down or becoming overweight quite uncomfortable, depending on the size of the chip. Putting it in the back could make sleeping harder.
I suppose everything comes down to the size of the chip, whether or not it has to be removed in order to be verified (could officials simply scan the chip through the skin with an electronic device?), and how visible it must remain at all times.
However, as a side note I'd like to point to two chip-ish instances in video games: The location of Cortana in Master Chief's head in Halo, and the bar code on the back of Hitman's head in Hitman. Depending on your own rules, this area of the head could be a viable place to put the chip. There's a lot of ways to play with that idea. Put a small disk tray like device back there, or force the populace to keep their heads shaved. Overall, I'd still say that this can be kind of a difficult question to answer with so many variables.
A skilled surgeon can remove and replace almost any single organ or tissue of the body, but the brain is what make you who you are.
Yes we do have a number of skilled neurosurgeons in the world, but they are few and far between; and even if you could convince a sufficiently skilled surgeon to do it, I think the downside (brain damage) would be a pretty good deterrent against removing the chip.
Putting a chip inside a child's brain would carry risk, but if performed correctly (which standardizing would likely ensure) should produce little to no damage.
The brain is surprisingly resilient in recovering from damage, as shown in people recovering from severe brain injuries, but who would want to go under surgery to remove a chip, knowing that the person waking up wasn't going to exactly be you.
