"I ask Mr. Tanaka to close the door."
In this case, と is a quotation particle. Think of it as quotation marks; the sentence basically says:
"to close the door", I ask Mr. Tanaka.
Especially in cases like this, I agree it can be a little confusing to English speakers. Here is a detailed explanation about quoting in Japanese.
I would even go further in your explanation, saying that the broken English translation could
"Close the door", i ask Mr. Tanaka.
This can explain the て form of しめて as a polite request (on top of explaining the と as the quotation marks). Otherwise, "to close the door", i ask Mr. Tanaka would be ドアをしめるとたのみます。 but I'm not sure that's a correct sentence...