Infinitives used as imperatives?

The quotation in your question omits two key items from the original:
1. that the quoted paragraph starts with the word "Memorandum:—"; and
2. that the following text immediately precedes your quotation:

I turned to the more interesting object of investigation which was presented to me by Rosanna Spearman's letter. The address was written as follows:--"For Franklin Blake, Esq. To be given into his own hands (and not to be trusted to any one else), by Lucy Yolland."

I broke the seal. The envelope contained a letter: and this, in its turn, contained a slip of paper. I read the letter first:

"Sir,—If you are curious to know the meaning of my behaviour to you, whilst you were staying in the house of my mistress, Lady Verinder, do what you are told to do in the memorandum enclosed with this—and do it without any person being present to overlook you. Your humble servant,"Rosanna Spearman."

I turned to the slip of paper next. Here is the literal copy of it, word for word:

Specifically, that text includes the instruction, "do what you are told to do in the memorandum enclosed with this ..."

That the list with the 'sentences' all beginning "To ..." is a list of instructions to be carried out by the recipient of the letter: that is why they all begin "To ...". In fact, the word "To" could have been omitted from the beginning of each instruction, such that the items in the list begin:

Go to the Shivering Sand at the turn of the tide.
Walk out on the South Spit, ....
Lay down on the rocks ... ...

With the word "To" omitted from each instruction, they are in the normal format for instructions:
"Do this ...; do that ...; etc."

So it's not a "poetic style"; nor misuse "of the infinitive"; nor "some other 'tense'" - but just a list to instructions!

Addendum:
A comment to my answer asked why the instructions start with "To do" rather than just "Do".

I have no 'definitive' answer to that. What I would point out is that the book dates back to the the 1870's: I wonder whether the form "To do ..." (rather than just "Do") was used at that time.

In any case, we do still talk of a "To Do list", and a "List of things to do", so (to my mind) there does seem to be a close connection there.