As soon as I realised the big box placed outside of my room, which had been making you feel inconvenient to go along the corridor during the past 3 days, now I would like to say sincerely sorry to you.
The period of the inconvenient time was happened 3 days ago till the time that I realised the problem. So is it grammatically correct to use the tense of past perfect continuous with the word "during" in this example?
It might be possible, but other problems with the sentence make it difficult to be sure. As soon as I realised . . . has normally to be followed by a main clause containing a verb in the past tense. It cannot be followed by the modal would. The whole sentence really needs recasting. The way you put it depends your relationship with your neighbour, but if the relationship is a formal one, you might want to say something like:
I’m very sorry if the large box outside my room has caused you any inconvenience over the past three days. I realise it must have made it difficult for you to get down the corridor, [if appropriate: but you’ll be pleased to know that I have now removed it.]
If you want to start the sentence with as soon as, you should rephrase it as follows, for example:
As soon as I realized the big box placed outside of my room caused you any inconvenience, I removed it. Now I would like to sincerely say I am sorry.
Realize is used as "I realized I was rude." In your sentence, taking off the parenthetical, you said "I realized the big box placed outside of my room"; it misses a verb. You should remove the comma and which, to obtain "As soon as I realised the big box placed outside of my room had been making you feel inconvenient to go along the corridor during the past 3 days." It is too verbose; "caused you any inconvenience" is a shorter way to say what you are trying to say.
"As soon as" is used as "As soon as she arrives we are ready to go." Your sentence is missing a clause. What did you do when you realized the box caused her/him any inconvenience?
You cannot use a comma before "now I would like to say sincerely sorry" since that is a new sentence.
