By making the a priori assumption that we are equally likely to be anywhere along the chronological list of humans, the Doomsday argument implies that our position n is correlated with the future total number of humans, N.
But if the many-worlds interpretation of quantum theory is correct then all futures will exist with all values of N. In that case our position n is not correlated with any particular future with a particular value of N.
Thus, as it seems reasonable that we cannot predict the future of the human race, can we use the failure of the Doomsday argument as evidence for the many-worlds interpretation?
Using Bayes's Theorem the probability of the total population N given our position n, P(N|n) is P(N|n) = P(n|N)P(N) / P(n).
If we assume prior complete ignorance of n and N then we should use the improper priors P(N)=1/N and P(n)=1/n.
Assuming a uniform probability for our position n given a particular total population size N, P(n|N)=1/N, we find that Bayes's theorem says
P(N|n) = n / N^2.
This is the Doomsday argument prediction. It implicitly assumes only one future with some particular total population N with prior probability P(N)=1/N.
But in the many-worlds scenario we would have many futures with many values of total population N weighted by the function W(N)=1/N. The probability of our position n would then be given by the sum
P(n) = Sum[N=n to infinity] P(n|N) W(N)
P(n) = Sum[N=n to infinity] 1/N * 1/N = 1 / n
As mentioned above the probability P(n)=1/n and W(N)=1/N implies total ignorance. Therefore in the many-worlds scenario if we lived until the end of the human race our birth position n would not tell us anything about the final population size N that any particular version of ourselves will experience.