Learn Hardcore Chinese: qǐng nǐ bǎ mén guānshàng ba.

把 marks a “disposal” construction: Subject + 把 + object + verb (+ result/complement). It foregrounds the affected object and a resulting change. In 请你把门关上吧, the door (门) is made to change state (from open to closed). The 上 after 关 shows the result “closed.”

Can I say it without 把?

Yes:

请你关上门吧。
请你关门吧。 All are natural. With 把, the sentence is more object-focused and slightly more “operation-like.” Without 把, it feels a bit more neutral. Meaning is basically the same in this case.

What does sentence-final 吧 do?

吧 softens a command into a suggestion/request. 请你把门关上 is fine but more direct; 请你把门关上吧 sounds kinder and more natural in conversation. It’s not a question particle here.

Why is 上 used in 关上? Does it mean “up”?

Here 上 is a resultative complement, not a literal direction. 关上 means “close (to the closed state).” Many verbs use 上 to indicate completion/attachment: 合上 (close/shut a book), 戴上 (put on), 系上 (fasten).

What’s the nuance difference among 关, 关上, 关掉, 关好, and 关严(实)?

关: general “close/turn off.” With doors, it can mean close, but can be ambiguous (e.g., 关门 also means “close for business”).
关上: close to the shut position (doors/windows). Very natural here.
关掉: “turn off” (lights, devices). Not for a physical door.
关好: close properly/securely.
关严 / 关严实: close tightly with no gap.

Where can 请 go? Is 你 required?

Common: 请你把门关上吧 or 请把门关上吧.
You can omit 你 if the addressee is obvious: 请把门关上吧.
Don’t say 你请把门关上吧 or 把门请你关上吧; those are unnatural.
For extra politeness: 请您把门关上吧 (using honorific 您).

Do I need a measure word before 门? Why isn’t it 那一扇门?

Measure words are needed when counting or specifying (this/that/how many). Here 门 is a definite, context-obvious object, so no classifier is needed. If you specify, use 扇 for doors: 请把那一扇门关上吧。

Can I add 一下 to sound softer? Where does it go?

Yes, use:

请你把门关一下吧。
请你关一下门吧。 Avoid 关上一下; 一下 usually follows the verb directly (without the resultative 上).

Where would 了 go, and does it belong here?

For a request, 了 is not necessary. 把门关上了 states a completed action (“[someone] closed it already”). As a milder command, 把门关了吧 is also common and acceptable, but 关上 emphasizes the closed state more clearly.

Is the pronunciation/tone anything tricky?

请你 undergoes third-tone sandhi: pronounce as qíng nǐ (请 → rising 2nd tone in speech).
把 is third tone (bǎ).
关上 is guān-shàng (上 is 4th tone). In fast speech, the complement can sound lighter, but dictionary tone is 4th.
吧 is neutral tone.

Is this polite enough? Any alternatives?

It’s polite. To soften further:

麻烦你把门关一下。 (Could you do me a favor and close it?)
能不能把门关上？ / 可以把门关上吗？ (Can you…?)
To seniors/strangers: 请您把门关上（吧）。

Can I drop the object and just say “Close it”?

Yes, if context is clear:

请关上吧。 (Close it, please.)
More natural with an object though: 请把门关上吧。

What’s the difference between 把门关上 and 把门关着?

关上: the action/result of closing (get it closed).
关着: continuous state “be closed.” As a request, 把门关着 means “keep the door closed (and leave it that way),” not “close it now.”

How do I make the negative command?

别把门关上。 (Don’t close the door.)
不要把门关上。 is also fine. For prohibition/ability: 不能把门关上。 (You mustn’t/can’t close it.)

How does word order change without 把?

With 把: 把门关上 (object before the verb).
Without 把: 关上门 (verb before object). Don’t say 把关上门; after 把, put the object right away.

What about the idiom 关门 meaning “close for business”? Is there ambiguity?

关门 by itself often means “(a shop) is closed.” In your sentence, adding 把…关上 makes it unambiguously “close the door.”

Can I say 把门给关上?

Yes, colloquial 给 can intensify: 把门给关上 (“go ahead and close the door already”). It can sound stronger or slightly pushy; use with care in polite contexts.

What’s the passive version?

门被他关上了。 (The door was closed by him.) For a state: 门关上了。 (The door is/has been closed.)

How do I say “I can’t close the door (it won’t shut)”?

门关不上。 (It won’t close; can’t get it to the closed state—mechanical/fit problem.)
关不了门。 (Cannot close the door—may imply permission/feasibility rather than the physical result.)