Is it a sin or forbidden to follow any of the parts of the Law of Moses? In John 7:53-8:11, Jesus condemns fulfilling the Law of Moses when it requires that a crowd stone an adulteress to death. This seems to imply (if these passages are authentic, which is in question) that in some cases fulfilling the Law of Moses is now sinful.

Even if this passage is not authentic, there are parts of the Law of Moses that now seem rather barbaric. These laws requiring the killing your children if they are stubborn and rebellious (Deuteronomy 21:18-21), killing a rape victim for not putting up sufficient resistance (Deuteronomy 22:23-24), and killing those who commit blasphemy or believe in a different religion (Leviticus 24:16; Deuteronomy 13:5-10 respectively).

Are there some cases where it is sinful or forbidden to follow any of the laws in the Old Testament, now that Jesus has fulfilled these laws (Romans 10:4; Galatians 3:23-25; Ephesians 2:15)?

A very good and very similar question is: To what extent does the Law of Moses still apply? and also:
Do we have to obey the laws of the bible? If so, what laws?
Even though these are questions that I'm interested in and still find somewhat confusing, in this question I'm looking for additional insights (scripture or scholarly) about whether it is now sinful or forbidden to follow any of the Laws of Moses (including the ceremonial laws for which we are no longer bound). Answers meant to address which of these laws are forbidden are better suited for these questions.
The topic of the law in relation to the Gospel is one of the most discussed ones in the entire history of theology. The space here is not sufficient for arguing one view as opposed to the other. I am going to answer your question with a simple "yes", though. Sometimes a contextless, literalistic application of Old Testament law is in fact sinful, and I am going to base that on three principles.


Rabbinic wisdom in the tradition of Hillel. When one law contradicts another in a specific circumstance, you must chose the most important one. E.g. when asked by Gestapo about the whereabouts of some hidden Jews, the moral thing is lying. In many instances Jesus aligns himself with this rabbinic school, as opposed to the tradition from Shammai.
Following that principle, there are further moral insights in the New Testament, that will on many occasions make a compliance with the old law an immoral act.
The law in the OT was given to serve many purposes. They include instructions on how to evaluate witnesses and administer punishment, including capital punishment. About this four sub-points spring to mind:


In comparison to other laws from the same era, e.g. the law of Hammurabi or the Greek law of Draco the laws of the OT are very humane. One is not to cut off limbs of thieves, or go beyond what is a reasonable punishment.
That contrast, IMO, has a value. The tendency is instructive, perhaps even more than the individual examples. In a society such as ours today, we had better follow the intent of the law, than literal words addressed to an early bronze age society. And that's not because I do not believe the words are true. I say that exactly because I believe they are.
Instructions about how to evaluate juridic guilt and punish offenders where not given to individuals, but to the people as a whole. Just as we today may not act as policemen, judges or prison wardens except in an official capacity, so this aspect of OT law never, ever were intended for anything but officially assigned people. And that includes the law "an eye for an eye". It is not about individual revenge. (Nor is it about inflicting physical damage, but that is another topic.)
In the New Testament focus has shifted away from how to organize a priestly state of a chosen people, to how to build communities, made up from all peoples, that reflect the Kingdom of God. That renders all aspects of the OT law, that concerned the treatment of suspected or convicted lawbreakers, void. It has served its purpose and while it was not wrong, it is not any longer applicable. Trying to apply it as if it were, would be contrary to the Gospel, and thus, yes, sinful