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?

"Relevant" and "irrelevant" are not well-defined statistical terms that everyone agrees on.
Case in point: the answer by @Demetri Pananos interprets "relevant" as "causal" and the answer by @Sextus Empiricus interprets "relevant" as "included in the model that we assume is the true — or at least a valid — model for the data". Both interpretations are meaningful but they are not equivalent.
Consider for example a (correctly specified) model for the response to a medical treatment vs placebo that includes gender and age effects: $E(Y) = \beta_a\textrm{age} + \beta_g\textrm{gender} + \beta_t\textrm{treatment}$. The effect of the treatment, $\beta_t$ is causal: it's the difference between the outcome if a patient is given the treatment vs if the patient is given a placebo. However the patient doesn't "cause" the outcome of his/her/their own treatment with personal biological and genetic characteristics. A better term for age and gender in this model is "covariates" or "effect modifiers". They are certainly relevant in order to prescribe each patient the most appropriate treatment.
Another popular attempt to specify what "relevant" means is to make a distinction between "practically significant" and "clinically significant" on one hand and "statistically significant" on the other hand. For example, an effect size might be statistically different from 0 but very small in magnitude  (absolute value), so perhaps not particularly useful in understanding phenomenon X, the argument goes.
Practical vs Statistical significance 
Is there a colloquial way of saying "small but significant"? 
There are subtleties with this interpretation of relevance has as well:
Stop talking about “statistical significance and practical significance”
So one takeaway is: If someone uses the term "(ir)relevant", "(in)significant", etc. to discuss their analysis, ask them "What exactly do you mean when you say your result is [a generic term hinting at importance]?"
