Question: I was wondering are there exist any restrictions as to what kind of research can be published. In particular, are there any laws in place that require certain published works to be removed from the public record?

It seems fairly odd that you see a dozen negative studies on acts which are illegal, but not a single positive one. It is known that the internet is filtered, so that might explain why you might not find positive studies using the popular search engines.

The only case I can think of would be homosexuality (where research all came out after legalisation).

Edit:
The question is regarding all studies conducted for cases which are illegal which includes but not limited to Class A drugs, paraphilias (positive study would be under sexual orientation but should not be limited to thought only) and incest. These all also have negative stigmatization so I would request if anyone is going to respond do not go on the base of you emotions, I'm attempting to test the validity of research on this case of bias and who can and cannot publish. One user mentioned 'prohibited' research, I was in the assumption that apart from conducting the illegal act you can still research on criminal cases even if it is for benefits or am I mistaken?

One case that questioned the whole research process is the FGM case. Where you have dozens of cases of it not showing any medical benefits, but their are people who have done studies showing or at least comparing to existing cases. However what I am more interested about where are all the studies that were conducted in the countries where it is legal, they exist but they are not available in the places where you find all other researches (for all the health organisation to make the claim that there are none)?

Answer: 
The existing answers already correctly state that there are of course things that you are not allowed to publish on, period - for example state secrets, or national security related research. Depending on how strict you are about the word "allowed" some industrial research that is covered by a Non-Disclosure Agreement may also fall under this umbrella.

That said, your wording and example (homosexuality) implies to me that you are more looking for some "thought control" type of laws, where you are allowed to publish about a specific topic, but your results can only point in one "acceptable" direction. This is not the case in modern democracies, although to some extent this certainly happened historically or in non-democratic societies.

In fact, people certainly do conduct research and publish papers that compare (for instance) the danger of legal and illegal drugs, and the result is not always that legal drugs are less unhealthy than illegal ones (in fact alcohol is often identified as a particularly damaging drug in such studies). Conducting such a study may require specific approval (if possession of some of the involved substances is illegal), but there is no law against publishing research that makes existing legislation look bad. In [Freakonomics](https://rads.stackoverflow.com/amzn/click/com/0060731338), the authors describe another example in detail, a study where an economist did embedded research in a gang to study their organisational management. Gang activities are certainly very illegal, but observing and reporting that gangs are actually quite well-managed from the point of view of management science is not.

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Consider that there are many reasons why studying something legal is much easier than something illegal. Access to study subjects is an obvious one - as long as homosexuality was illegal it is hard to identify study participants and to convince them to participate in a study. Funding for and public interest in studying activities that are illegal is typically also much lower (unless the research is specifically about preventing the illegal thing from happening). Finally, there is a certain amount of self-censorship embedded in all human societies - sure, many academics have tenure and could in principle study whatever they want, but in practice few people will voluntarily choose a touchy study subject that has the potential to make them pariahs.
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> are there...any restrictions [on] what...research can be published. 
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Yes. For example, national security laws restrict the publication of certain works and legality of human/animal experimentation restrict others.