In-class, allowing students to address professor informally ('tutoyer' / 'tutear') Supposing that Dr. A is teaching an undergraduate-level course, what would be the pros and cons for Dr. A to allow students to address himself in an "informal" manner? 

By "informal" here I mean using what is employed in a number of languages for informal conversation. For example, this would be tutoyer in French or tutear in Spanish. (Hence, the question does not apply for an English-speaking classroom, but does apply for a French or Spanish-speaking one).

EDIT (clarification, thanks j91): In Spanish, French, German and other languages, the second person singular has two versions: "Tú", "Tu" and "Du" are the informal versions in Spanish, French and German, respectively, and "Usted", "vous" and "Sie" are the formal ones in the same order. "Tutear" is the act of using systematicaly the informal version and, in theory, should be avoided when speaking to a person in a position of authority.
Your question is completely subjective and context dependent. It adresses the balance between formal and informal, which depends on culture, personality traits, position in the academic hierarchy, etc. It's like asking, which are the pros and cons of teaching in full business suit vs Hawaiian shirt?

Some students will feel more confident if the class environment is informal. Some won't. Some won't see the difference. You (assuming you are the Professor) may feel at ease being talked in an informal tone, or maybe not. If you are in a new country or university and need to figure out your way, attend some classes and check which is the standard. Nonetheless, whatever you chose, don't withdraw your authority as Professor.