I'm not by any measure an intellectual capable of (armed with) all the details for "refuting" his position.
Read my posts.
Christopher Timpson did an undergrad thesis that put Deutsch's FOR in perspective. I know of no NO known, public exchange of ideas on the content of the remarks. David seems to be considered by many as some kind of "genius", which in quantum physics, he seems to have a good knowledge. Nevertheless, MANY critiques exist of his positions concerning the interpretation of qphys called Many Worlds. Cambridge published a book last year - very technical discussions in PhiSci between a couple philosophers and many of the MW - multiverse people.
It really doesn't elucidate much except the very technical subjects tied to how Everrettian thesis explain probability coherently - within the terms of Everrett's and deWitt's positions.
What you also won't find is any public record of intellectual debate between the Stapp fans and their derivative positions and those of the MWI. I don't know why, but I have heard that or instance, Robert Penrose is very very opposed to the fundamentals of Deutsch's FOR. Hearsay? I donna know but their clan - on the FOR user group - won't let you talk about subjects like "mind" - it just gets moderated out, nor any questions of philosophy of religion and issues with the implicit teleology of Chapter 14 of FOR. The only book taking that up, which no one on the user list wanted to talk about,was CRITIQUE OF THEOLOGICAL REASON, a kind of Kantian approach to analyzing the key structures to various kinds of theological presuppositions and their consequences.
Idem for Oppy. He has done books about multiple worlds, arguing about the notion of "gods", "infinity", and a bunch of other subjects, all within the perspective of David Lewis' ideas on possible worlds.
So, critiques, you won't find many. Christopher Norris does 3+ chapters in QUANTUM THEORY AND THE FLIGHT FROM REALITY in which he claims that the ideas that require use to imagine multiple worlds, etc, are not necessary, and that Deutsch's arch-rival, David Bohm, is a better way of understanding quantum theory interpretations.
Take your pick. You won't find much.
However, start by reading on the subjects of "explanation" in phiSci, all the auxiliary subjects related to the question of epistemology in quantum theory, as well as the subjects in areas such as complexity theory.
Again, there is a problem with the basic suppositions. Knowledge and its creation, advancement is seen as following a linear like progression similar to the simplistic idea that time is like an "arrow" - linear.
there are ALOT of ideas about all this and frankly, rather than David using his genius to discuss in a general manner more & more new subjects (the physics of Beauty, Goodness) he could have clarified his ideas in FOR and gone on to develop new perspectives on how the multiverse works (read FOR).
All we get are anecdotes, allegory of Hilbert's hotel and infinity spaces. This is all better explained and in a more detailed, tradition, scholarly manner, elsewhere.
I consider David as a "cursor" pointing the way for certain subjects that were, often, monnaie d'échange (currency) in the 50-60-70's amongst the philosophers in David K Lewis circle (Australians mainly) re-read from the point of view of a rather exotic set of interpretations part of the Oxford school, Many Worlds theory (Alan Wallace, Simon Saunders, etc).
Timpson refined his positions and published them in a book of essays on the uses of Turings ideas. Excellent rewrite - but his undergrad thesis was right on for certain subjects.
moreover, you see in TImpson's critique, that he has identified Deutsch's Popperian tilt, and its implications, as well as his FOR thesis having circular conclusions if FOR's epistemology is read as a naturalized epistemology.
Basically the idea coming from Bennett at IBm and his forefathers, that "information" is some kind of exotic "physical" substantial stuff, is completely demolished.
Another physician took defense of Timpson and brought up the importance of Bayesian considerations in the interpretation of Qphys and the measurement problems.
All that has limited readership but the papers are on XARCHIV.ORG if you do some Googling.
Keep in mind. Though Popper has an interesting, highly respected approach, he is not all that mainstream, and especially the idea of "evolutionary epistemology". There are some highly technical books on that, as well as the idea that "mind" coevolves with physical-biological forms following some sort of "rules" - perhaps Darwinian.
Yet again, not everyone is in agreed (to say the least !) on the INTERPRETATIONS of Darwinians rather limited ideas written over a century & half ago ! This is crucial because if ever (and it will NECESSARILY HAVE TO OCCUR in a Popperian perspective of the evolution of knowledge) "Darwinian" is usurped, and the theories changed radically enough to make many of the expository mechanisms in David's BOI arguements fail, then his work will be left with a skeletan of "hey, be positive ! It's much better than being negative, which is bad. Bad explanations make us not see beauty - tantamount to a physical incarnation of a kind of "truth" in things) and Good explanations reveal enough of the mystery to contine the quest for INFINITE KNOWLEDGE.
Well. If I don't agree, and I am part of those that DONT AGREE with the naive form of the BOI ideas, then they will disappear, mutate, and paradoxically they have proven to be true in becoming false.