I work at Intel and have worked on reliability for server CPU's. (since talk is cheap on the internet, my Intel email is in my profile, you can email me at Intel if you don't believe me)
I agree with what FrankSchwab, GammaLaser, exdeath, rahgu and CanOworms all wrote.
So, yes, most of the time a single transistor failure anywhere on the CPU will result in the entire CPU not working - because the transistors that would tend to fail would be transistors that are used a lot and thus a failure in one of these commonly used transistors (or wires) would result in the whole thing not working. As FrankSchwab wrote, sometimes this failure will be immediate and catastrophic, other times it might be more of a flaky failure.
There is redundancy in server CPU's - register files use ECC and can fix failures on-the-fly, Intel uses a technology called "Cache Safe Technology" to detect cache regions with too many ECC errors and can disable that region and then swap in a redundant region on the fly, and there are other redundancies. There are also lots of transistors on a CPU that are used for test or debug that have no effect on the operation of the CPU. Or you could have a bit get stuck high (or low) in something the LRU (least-recently used) circuitry for a cache, or the branch prediction circuitry such that the CPU would suffer a performance issue but otherwise continue to work fine.
But most of the time, when I look at CPU that doesn't pass the test screen (and one of these performance issues would be caught), I tend to see it's just one transistor that failed. And generally it's a specific lithographic mask structure that fails... like you'll get something where there's all metal 1 in a square, with a via (a vertical wire) through the middle of it, and there will be a transcription error in the fab where the via mask and the wire mask didn't totally line up resulting in via that's much narrower than it's supposed to be. And you'll see this particular structure show up all the time in the failures list, and then you can go back, search through the whole design and just fix every single one of these errors on the whole CPU (for example by making the via slightly bigger). And then you'll see a huge improvement in reiability and yield by fixing just this one issue (that happened thousands of times all over the whole die).
To be honest though, my thinking that any one failure generally results in a CPU failure could be a form of sampling bias. Because the failures that I've looked are like this, I then think that they are all like this. But if a transistor fails in an unused chunk of circuitry, would anyone ever know?
I will say, in response to ArchAngel777, that in my experience there's a surprising lack of redundancy in desktop and consumer CPU's. Modern CMOS process technologies produce transistors that generally work for a really long time (ignoring early failures caught in burn-in), and so redundancy doesn't do have much function in real life and burns power, and take up space. So, no, there's an amazing lack of redundancy in most consumer oriented CPU's.