Is $i$ irrational?

It depends on convention whether or not the scope of "irrational" includes non-real numbers. As you say, if the irrationals are defined as $\,\Bbb R\setminus \Bbb Q\,$ then $\,i\,$ is neither irrational nor rational. However, many authors use "irrational" to mean "not rational", i.e. $\,\not\in \Bbb Q,\,$ therefore $\,i\,$ is irrational. This is quite common usage in university-level algebra.

For example, in my 2006/3/8 sci.math post I remarked that if one searches books.google.com for "irrational algebraic" one finds such usage by many eminent mathematicians: e.g. John Conway, Gelfond, Manin, Ribenboim, Shafarevich, Waldschmidt (esp. in diophantine approximation, e.g. Thue-Siegel-Roth theorem, Gelfond-Schneider theorem, etc). See also other posts in that sci.math thread titled "Is $\,i\,$ irrational"?

Our of curiosity, I ran another Google Books search on "real irrational" numbers. Authors using such terminology presumably employ the more general definition of irrational numbers. Among such authors I found the following eminent mathematicians: Bombieri, Davenport, Dedekind, Euler, Hurwitz, Kronecker, Kirilov, Mahler, Lang, Ostrowski, Ribenboim, Weil.

However, it is not easy to find an explicit definition since higher-level textbooks assume the reader already knows basic terminology. I vaguely recall that someone (Gerry Myerson?) once posted to sci.math some links to definitions which make it unquestionably clear that the author employs the more general definition of "irrational". Perhaps someone can dig those up, or locate others.

In any case, it is a matter of convention. In most cases one can quickly infer the intended denotation from the context, so there is little chance for confusion.

It is neither. Just as it's neither positive nor negative. The real numbers are partitioned into rational and irrational numbers; but things which aren't real numbers don't have to be one or the other.

Your conundrum is similar to saying "A $2 \times 2$ matrix is not rational. So it must be irrational. But it's not irrational; a contradiction." Does it seem more clear why that's a false dichotomy?

If I tell you that for lunch you can either have pizza or pasta, and you see someone eating a burger, does that negate my statement? Or reality?

Context is not absolute in mathematics, and it's important to remember that. It is true that in the context of the real numbers we have rationals and irrationals, and whatever is not rational is irrational. But in the context of the real numbers we also don't have $\sqrt{-1}$.

In the context of $\Bbb C$ we often talk less about irrational numbers so the context can be taken to both direction. It is possible to declare "In the context of the complex numbers, the irrational numbers are $\Bbb{C\setminus Q}$" in which case $i$ is certainly irrational, but it is also possible to make other declarations, like "The irrational numbers are still $\Bbb{R\setminus Q}$", in which case $i$ is not irrational -- but it doesn't make it rational either.

It's a question of context and it's not true that there's always a conventional and agreed upon context. This very question is a good example for that.