Answer
No, $\frac{1}{0}$ is not an irrational number.
Work Step by Step
Letting $R(x)$ denote "x is rational" and $I(x)$ "x is irrational, we have the following definition of the relationship between rational and irrational numbers: $\forall x\in\mathbb{R}(\neg R(x)\Rightarrow I(x))$, i.e., for all real numbers $x$, if $x$ is not rational, then it is irrational. But $\frac{1}{0}$ is not even in the domain of discourse, the real numbers, so we cannot use the definition to deduce that it is irrational.