If it is written that X is not as tall as Y, it indicates that X and Y have unequal heights. But does this necessarily imply that X is shorter than Y?
In the absence of further elaboration, yes: "I am not as X as you" implies that you have more X-quality than I do. If X is "tall", then I am shorter than you.
It is possible to follow up for a humorous (or insulting) reversal of the normal implication:
I am not as good as he is... I'm actually way better.
But if you don't do this explicit countering, the default reading is that "not as X as" is always "less X than".
Logically, I do not think it should imply X is shorter than Y.
The definition of "as" according to Dictionary.com is
to the same degree, amount, or extent; similarly; equally
So, if I were to say "X is as tall as Y." I would be saying that X is equally as tall as Y.
If I were to negate that statement, then I would have "X is not as tall as Y." Which would be saying that X is not equally as tall as Y.
However, in general speech, I do agree with Hellion that it implies X is shorter than Y.
I see from the comments and downvotes that I have to spell it out completely for some readers.
First, if X is not as tall as Y, we can say that one of them is taller; that X is not equal to Y. Let us first assume for the sake of argument that Y is the taller, meaning that X < Y. My original comment read:
X < Y has the inescapable corollary that Y > X. As posed, the question needs no further answer.
I had assumed (it seems wrongly) that readers would appreciate that there is an equally logical mirror statement:
X > Y has the inescapable corollary that Y > X.