Implicit gender bias is present in academia (at least in the sciences). I've heard that some journals hardly accept any manuscripts written by Muslims. To what extent do the religious beliefs of an author affect their paper's chance of being accepted? Have there been any studies that look at implicit religious biases?
In my entire scientific career, I have seen religious belief come up precisely once: an (apparently Christian) author closed a paper with "AMDG," which apparently stands for Ad maiorem Dei gloriam. Of the set of three peer reviewers, one ignored it, one was confused by it, and one asked for it to be deleted as irrelevant to the substance of the paper. The authors removed the acronym as part of their revision (amongst other changes) and the paper was published.
The other place that I know that religion comes into play in scientific publication (though I have not personally encountered it), is when a person attempts to use statements from their religion as evidence for a position. This never goes well: in a scientific context, claiming "the Bible says ..." or "the Koran says ..." is equivalent to saying "The Lord of the Rings says ..." or "Fifty Shades of Grey says ..." Such a document is a secondary source at best, and one that has no particular reason to be regarded as a reliable source of evidence, despite the magnitude of cultural belief. Such books can, of course, be cited and analyzed just as any other literary/historical work, but that is different than using them as evidence for a scientific position.
So, in short: if you don't make your religion an issue in your manuscript, it should never come up as an issue in the normal scientific publication process. This doesn't mean you might not encounter horrible and inappropriate behavior (scientists are just as capable of being terrible people as anybody else) but it is definitely outside the norm of the peer review and publication process.
I want to address part of the question, because I think there's an important comment on the nature of such accusations.
The "this [bad thing that happened] was because [person or group X] is prejudiced against [group Y with which I'm affiliated]" trope is one that I've heard for many, many years, in part because I am part of multiple such groups Y which have been the subject of societal prejudice.
However, it is also the case that most of the people I've heard who have used that line of reasoning are using it as a rationale to place blame on someone else. Before jumping to claims of bias and prejudice, one should check that one's own behavior isn't at least partially the problem. For instance, blaming prejudice for not getting a job when one's cover letters are pro forma templates and one's CV is full of typos and useless information is just shifting blame and responsibility. It's easy to do, and it perhaps makes one feel better about oneself ("it's not me, it's them"). But in the long run, it's a dangerous attitude to have, because it hampers one from the self-improvement needed to break out of the cycle.
It's plausible that there's cultural bias that's correlated with religion, but it's hard to imagine that religion is actually the primary factor. After all, referees typically have at best weak indications of the author's religious beliefs. Furthermore, cultural and religious bias could play out identically in practice, which makes them near-impossible to distinguish.
I don't think gender bias is a particularly illuminating comparison. There are widespread implicit associations of men with science and math competence, which at a nationwide level are correlated with differences in achievement. (See, for example, this study.) These implicit biases could also interfere with the evaluation of academic papers. However, whatever negative associations people have with Muslims, I don't believe being bad at science and math is generally one of them. That makes it a rather different form of bias, and one that's less likely to play an insidious role in evaluating academic work.
I've heard that some journals hardly accept any manuscripts written by Muslims.
This sounds like utter nonsense, although of course it can't be disproved. For one thing, there's no official list of journals: anyone could set up a web page entitled "Journal of Hateful Bigotry: No Muslims Allowed". Furthermore, I can imagine weird exceptions, such as theology journals run by religious groups that have no interest in an outside perspective. However, I don't believe any mainstream journal holds articles by Muslims to a much higher standard than other submissions. Certainly not in mathematics or related areas (which are the cases I'm most familiar with), and I'd be amazed if there were examples in other fields.
It's plausible that there's bias against researchers working in developing countries, which for researchers in developing countries that are primarily Muslim could be difficult to distinguish from anti-Muslim bias. This bias could manifest itself in various ways. For example, prejudice against topics that are particularly popular in developing countries, or against researchers without prestigious affiliations. There's no doubt that research from or on developing countries is underrepresented in the top journals in at least some academic fields, such as economics, but it's difficult to pin down exactly why. (One key difficulty is that we have no objective, absolute standards to compare with.)
I've never heard of religious bias existing in academia. You might face problems if your field is closely related to religion and you are expressing views that are fanatic and not based on clear reasoning or evidence. However, apart from that, I don't see any reason why journals would not accept papers written by members of a particular community.
However, I have heard of a case where an author's paper was taking a long time to be processed after acceptance by the journal because he was from a country that's politically unstable. According to this author, the journal required some special permission from the government for publishing papers from this region. However, all this was being taken care of by the journal; the author just had to wait a month or two longer than usual for the procedures to complete.