закрыть or закрывать?

Are the verbs used coorectly in the following dialog?

A:Нужно закыть дверь.
B: (Silence)
A: Нужно закыть.
B: (Silence(.
A: Закрывать!

It's not clear what this dialogue is about.
You missed some letters in the verbs.

A:Нужно закрыть дверь.
B: (Silence)
A: Нужно закрыть.
B: (Silence(.
A: Закрывай!

The last is the command form.

I am sorry for my incorrectness and unclearness.

The dialog should be as follows:

A:Нужно закрыть дверь.
B: (Silence)
A: Нужно закрыть.
B: (Silence)
A: Закрывать!

A tells B to shut the door twice but B does not react so A emphasizes the command by using the imperfective infinitive.
Is the dialog correct grammatically?

Yes, it is correct grammatically but Russians will hardly use imperfective infinitive for this verb in this situation. It can be misunderstood.

Use imperative (either perfective or imperfective) for the command here: "Закрой дверь!" or "Закрывай!"

Or, if you want to be rude, use past perfective for this purpose: "Дверь закрыл!"