How to be sure services and researches offered by the University are not becoming cases of unfair competition? I work in Italy on a research group working on the conservation of cultural heritage analyzing artworks. I had some discussion with my lab on what job or project the university shouldn't accept to avoid overlapping with the private market as unfair competition.

For instance, in my field, many papers are written as a case study where artworks are analyzed by the laboratory with conventional techniques and procedures and the results are published. In fact, there are small companies and freelancers that provided the same service and I think this is a case where the university might be guilty of unfair competition because it offers the same service for free. Of course some times there isn't the money to perform the service while the artwork could benefit from this research for their conservation so this practice sometimes could have a positive impact on the society.

I am aware that in this specific case we are not committing any crime but I wanted to reason about the role of the university.
Are there any guidelines, or law related to the role of the university and what is considered unfair competition? Where I can ask for clarification?

This is not restricted to my particular field because I've found the same problems also dealing with other disciplines.
I think your assumption that the university and private evaluators "offer the same service" is flawed. A private analysis will be delivered to a client and not to the general public. Being private, it may give some competitive advantage and so has monetary value. The university, on the other hand provides an information "service" to the general public so that all can benefit. 

To "forbid" the university from doing some sort of research so that the possible results could be locked up for private rather than general benefit seems to be a pernicious idea. 

And, the "service" of the university isn't actually free (as in beer). Resources go in to it that are paid in a variety of ways, sometimes from public monies since the public benefits. But the information it provides is free (as in speech). 



But for your question about law, note that laws vary from place to place, and not all laws are wise.