In what chapter of The Tenant of Wildfell Hall by Anne Brontë, does the story shift from past tense to present tense, and why does it shift?

Perhaps your question is worded a little wrongly. Chapter I starts out in present tense but does not stay that way long. The long flashback of the chapter is told in past tense. The very last paragraph of the chapter does, however, return to present tense, showing that Gilbert is writing a letter (i.e., an epistle) and continues for the first few lines of Chapter II.

You must go back with me to the autumn of 1827.
My father, as you know, was a sort of gentleman farmer
[...]
Now, Halford, I bid you adieu for the present.  This is the first instalment of my debt. ....
Yours immutably,
Gilbert Markham.
CHAPTER II
I perceive, with joy, my most valued friend, that the cloud of your displeasure has passed away; .... (Ch. I & II)

In Chapter XVI (16), Gilbert ends his past tense flashback narrative with a present tense comment, then switches to transcribing Helen Graham's diary. Her diary is mostly in past tense as well as it records events of the day that happened in the immediate past (unlike Gilbert's flashback to distant past events). Yet, Helen does make present tense comments about herself as she records her life experiences:

I am not yet settled, and feel as if I never should be. ... I am quite ashamed of my new-sprung distaste for country life. .... (Ch. XVI)

Chapter XXVIII (28), a diary entry dated "December 25th," is a good example of the continual switches between present tense and past tense diary entries. It continues for a goodly length in a reflective present tense before moving to past tense accounts of events:

Now I am a wife: my bliss is sobered, ... Early in spring he announced his intention of going to London: his affairs there demanded his attendance, he said, .... (Ch. XXVIII)

The story ends with Gilbert reclaiming the epistolary narrative told to Halford in present tense as his flashback explanation of his romance is brought to a close: "But it is time to bring my narrative to a close." Since present tense and past tense are laced throughout the story, it is incorrect to ask at which one chapter there is a single shift. What is correct to ask is (1) at which chapter does Gilbert's flashback shift to Helen's diary and (2) which chapters have one or both past and present tense.

Gilbert's flashback shifts to Helen's diary at Chapter XVI. Many chapters have both present and past tense, such as Chapters I and II, Chapter XVI, Chapter XXVIII, and Chapter LIII. The shifts occur because and when the two narrators, Helen and Gilbert, shift from a present moment comment to a near-past or distant-past flashback or shift from a flashback to a present moment as recorded in a diary (Helen) or in a letter (Gilbert).