This answer purely concerns US law.
Your question asks one to determine if introducing evidence in violation of the rules of procedure is a criminal act. In order to fully understand an answer to this question, you need to understand a bit of background on how the rules of evidence.
So the rules of evidence are provided to create a fair trial. Over time, we, as a society decided that some type of evidence is inherently untrustworthy and that we should not base decisions off this evidence (likewise we added many exceptions that tend to make the evidence more trustworthy). Now in court in order for one to guarantee enforcement of these rules, one needs to make an objection. No person may object except for a person party to a case (an exception is if a witness is privileged from testifying).
Let's go over both those terms. Object means to state a formal grievance that a particular rule has been violated. If a party does not make an objection they waive the right to appeal on the violation of that rule and testimony that may be against the rules can be put before the jury (or other factfinder). Keep in mind that a jury will see you objecting and doing so excessively may look unfavorably like you're hiding information from the jury.
Let's discuss privilege. Privilege allows someone to not testify on some matter due to a socially recognized exemption from testifying. Some of these exemptions are doctor-patient confidentiality, attorney client confidentiality, right to not incriminate yourself, and in some states rights that sexual assault victims have.
With this all in mind we can begin to explore the issue of your question. Rules of evidence must only be enforced if a party makes an objection; however a judge may decide that a particular rule is very important and enforce the rule sua sponte or "on their own accord"; which means that although no side objected, the judge still kept that evidence out.
Now let's begin dissecting your question. Who are the parties to a criminal case? The State (in the form of a prosecutor) and the defendant. If the state begins to violate the rules of evidence the defense objects, makes a motion for mistrial and if they can prove the prosecutor broke the rules intentionally the person goes free due to double jeopardy. Will there be criminal charges filed? No it's not against the law to lose a case. That particular prosecutor may be fired.
Now let's look at the defense violating the rules. The prosecutor now must choose to object or not object. If they object the jury sees this as hiding evidence from them (and good defense attorneys exploit this) if they don't object than this evidence is allowed. If the prosecutor does end up objecting, the judge may disallow this inadmissible evidence and tell the councilor to move on. If the councilor continues to ignore the judge and to try to introduce the evidence, they may be held in contempt of court. According to Findlaw:
Civil contempt often involves the failure of someone to comply with a court order. Judges use civil contempt sanctions to coerce such a person into complying with a court order the person has violated.
Criminal contempt charges, on the other hand are punitive, meaning they serve to deter future acts of contempt by punishing the offender no matter what happens in the underlying proceeding. Someone incarcerated for criminal contempt cannot secure their own release by deciding to comply with the court.
Essentially criminal contempt is a much more serious version of civil contempt. A judge may find an individual in criminal contempt for flagrant disregard for the rules of evidence, but there will be sufficient warning from the court prior to this and their would have to be an objection of some sort or the court would have to enforce the rule sua sponte.
TL;DR and In the end, criminal contempt can be a result of breaking the rules of evidence, but it isn't automatic and there must be an order from the judge to not present the evidence first.