Can an irrelevant variable be significant in a regression model? I'm reflecting about statistical significance and I think a variable can be irrelevant and significant at the same time. But, how do I explain that? And if that were the case, should I include that variable to my regression model?

Here is a good example.  Suppose you are interested in modelling the effect of ice cream sales on incidence of shark attacks.  Now, clearly there is no association; buying ice cream in no way affects the incidence of shark attacks.  However there is a third variable which affects both, namely the temperature outside.
On hot days, people will want ice cream and might also want to go swimming.  Hence, hot days see increases in both ice cream sales and shark attacks (by virtue of more people going swimming and hence being at risk for an attack). This is known as confounding.
So clearly, ice cream sales is irrelevant when studying shark attacks but were we to regress shark attack numbers on ice cream sales we would find a significant result.  However, that statistical significance is confounded by temperature, and so the result is meaningless.
EDIT: Because this has generated conversation around the interpretation of "irrelevant" I feel the need to make some additional comments and concede I should have said "it depends" (even though I object to the arguments made).  If "relevant" is understood to mean "closely connected or appropriate to what is being done or considered" then "irrelevant" should be taken to mean "not closely connected or appropriate...".  In this case, ice cream sales could be considered "relevant" since they would be correlated -- but I find interpreting "relevant" in a purely statistical sense to be an extremely narrow (and irregular) way to attach meaning to "relevant".  None the less, it is an interpretation one may have.
If, however, you interpret relevance or "closely connected..." in a mechanistic sense -- i.e. how closely connected are ice cream sales and shark attacks in the sense that I could change the former and affect the latter or vice versa -- then ice cream sales would be considered irrelevant and my original comment applies.
