Learn English – Can the past tense be used with “now”

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.]

As @snailboat said in comments:

Shifting to the present tense in the middle of a narrative is a use of the so-called "historical present". It has the effect of making a story more vivid, more immediate. I think if you search the site for that term you'll be able to find some more detailed descriptions of it.

Most stories are set in the past, and as such are properly spoken in past tense. However, speaking of the past in present tense can indicate that the speaker's recollection is so vivid that the speaker almost feels transported back to the moment of the events. Speaking in present tense is also common trick in storytelling to a create greater sense of immediacy for the listener, in hopes that the listener will imagine the events in present tense, as if the listener were a witness to the events as they are happening.