Why does "Please approve it" sound wrong?

Whenever I read an email like this, the English sounds incorrect to me.

"I would like to take tomorrow off. Please approve it."

I want to say that "Please approve" is more natural, but why is that?

Of course, I can't be sure why it sounds wrong to you, but to me it sounds a bit funny because I'm expecting "it" to refer back to something mentioned previously, and the only thing in the previous sentence it seems it could refer to is "tomorrow". And asking someone to "approve tomorrow" is just silly — there will be a tomorrow whether they approve it or not.

A minimal fix would be either to drop the "it", as you suggest — which would make the sentence somewhat terse and elliptical, but which could work in a workplace e-mail — or to replace it with "this", which at least could validly refer to the entire request. But just to avoid ambiguity, I'd prefer to expand it all the way to "this request", as suggested by trpt4him.

Or, alternatively, the request could be rephrased e.g. as:

"I would like to take a day off tomorrow. Please approve it."

where "it" can now refer to "a day off" (viewed as a set phrase), which can be legitimately approved.

(Ps. I make no comment on whether the brevity of the request is appropriate here, since that doesn't seem to be what you're asking about and since I'm not familiar with your local workplace communications style. In some places and contexts, your example request might be exactly as long as it needs to be; in others, you might have to expand it to five times its current length to make sure it sounds polite and respectful enough.)