The stuff in this post is surely in the literature, but it's more fun to think it through oneself than to dig in the literature.
The Doomsday Argument holds that our position in the birth-order of humanity supports hypotheses on which we are roughly in the middle of the birth-order over hypotheses on which we occupy a more extreme position in the birth-order. Here's a back of the envelope calculation. Let's say that about 100 billion people have ever lived. Then we should expect that only about 100 billion people will live. If population stabilizes at 10 billion, that means that the earth's population will probably turn over only about ten times, and hence the human race will die out within 1000 years (life expectancy might go up to 100 years).
As a proxy for the complications of the full Doomsday Argument, consider a simplified version. The Demiurge flips a fair coin. If it's heads, he creates a single person. If it's tails, he creates a sequence of 100 people, a new person every 20 years. You come into existence, learn these facts from the Demiurge, and also learn that you're number one in humanity's birth order. By the Doomsday Argument, this should give you reason to prefer the hypothesis that the Demiurge got heads on his flip. And indeed there is a very nice Bayesian rendering of this. The probability that you're first given heads is one. The probability that you're first given tails is 1/100, since you could equally well have been anywhere in the sequence. So your position in the sequence strong supports heads. Indeed, if the prior probability for heads is 1/2, Bayes' theorem says your final probability for heads is about 0.99.
Birth order complicates intuitions a little. There are worries of essentiality of origins: could you have been anywhere else in the birth order? There is some reason to think that such considerations are irrelevant as what matters is epistemic and not alethic modality, but to rule them out, let's vary the case a little. The Demiurge tosses the coin. On heads, he creates one person on one planet. On tails, he creates 100 people, each on a different planet. And then after creating each person he takes a deck of cards numbered from 1 to the number of people (i.e., a deck containing just a single card labeled 1, or a deck of cards from 1 to 100), and gives one card to each person. You are apprised of all these facts and you see that you have a card labeled 1. You think: getting card 1 is very unlikely on tails, but is certain on heads, so probably the Demiurge tossed heads. The cards obviously correspond to birth order. It is clear that if the Doomsday Argument is sound, this argument is inductively good. The reasoning is exactly parallel.
But now consider a firing squad variant. One hundred people, including you, are blindfolded, and a different crack shot in the squad is assigned to each. The dictator tosses a coin. If it's heads, one of the shooters loads his gun with a blank and the others load a bullet. If it's tails, they all load their guns with blanks. They fire. You notice you're still alive. Then the dictator takes a deck of cards, numbered from 1 to the number of survivors, and hands them out randomly to the survivors. You look at your card before you look around for other survivors--perhaps smoke swirls around you and you can't see further than your card--and it has the number one on it. The exact parallel to the Doomsday Argument then would say that you have very strong evidence that the dictator tossed heads, since it would be unlikely that you'd see the number one card if there were 100 survivors.
But notice that if that's how you're thinking, you're neglecting a crucial piece of evidence: your survival. Yes, the number one card favors the heads hypothesis. But the earlier observation that you survived favors the tails hypothesis. Your probability of survival on heads is 1/100, and on tails it is 1. In fact, the two pieces of evidence cancel out perfectly. For consider the combined piece of evidence: you survive and get the number one card. The probability of this on heads is: (1/100)(1) = 1/100. The probability of this on tails is: (1)(1/100) = 1/100. So the overall evidence is neutral between the two hypotheses. Your probability of heads remains 1/2 as before the experiment.
If the Demiurge case with the deck of cards is analogous to the firing squad case, then in the Demiurge case we should say that overall you have no evidence for heads or tails. The diagnosis of what went wrong in the reasoning about the Demiurge case then is that the evidence of your existence was neglected. If it were taken into account, it would have canceled out with the evidence of the card number. And by the same token, in the Demiurge and birth order case, the evidence of your existence and the position in the birth order cancel out. And hence by parallel the Doomsday Argument fails.
Is the firing squad case analogous to the Demiurge case? The one potentially relevant difference between the two cases is that the firing squad case is about survival while the Demiurge case is about coming into existence. Does that matter? I don't know. But whether it matters or not, I do actually have an intuition that in the Demiurge case we should take into account the evidence of your existence. On tails, intuitively, you have 100 times greater chance of coming into existence. That precisely cancels out with the fact that on tails any one of the persons who comes into existence has a 1/100 chance of getting the number one card. And hence on balance no evidence is provided.