|Mary Tudor does not approve of your old timey language usage. (Via.)|
You might think this doesn't matter, but if your reader happens to be someone who has read a lot of Shakespeare or those other guys from the Renaissance, she might have absorbed correct thee/thou/thy/thine usage on a subconscious level. In this case, your clumsy attempt to sound old-timey will rocket her right out of your story as surely as a comma splice, using the word "exhort" when you mean "exert," or any other grammatical shoddiness.
So, here's how to use "thy," "thine," "thee," and "thou" correctly, with examples from Shakespeare.
"Thy" means "your." Here's a quote from Cymbeline Act 4 Scene 1. The speaker is Cloten:
Posthumus, thy head, which is now growing upon thy shoulders, shall within this hour be off.
"Thine" is a little bit tricky. It can mean "your" or "yours."
Polonius's famous speech to Laertes, from Hamlet Act 1 Scene 3 includes this line:
This above all, to thine own self be true,
And it must follow, as the night the day,
Thou canst not then be false to any man.
Here, Polonius is using "thine" to stand in for "your." Note that "thine" is sometimes (but not always) preferred if the word following begins with a vowel sound. So we have "thine own," but in Cloten's speech above, "thy head." The "n" sound in "thine" closes the word off so you don't end up with vowel sound soup.
"Thine" means "yours" in other contexts. In Sonnet 40, Shakespeare writes,
Take all my loves, my love, yea, take them all;
What hast thou then more than thou hadst before?
No love, my love, that thou mayst true love call;
All mine was thine before thou hadst this more.
You can't go wrong with "thy" and "thine" if you use "thy" to mean "your" and "thine" to mean "yours." You can also use "thine" to mean "your" if you want. So long as you never try to use "thy" to mean "you" or "thine" to mean "you," you will be fine.
"Thou" vs. "Thee"
In our modern English usage of today, we have one word, "you," that serves many purposes. It is the second person pronoun, used to refer to any number of people whom one is addressing, no matter where "you" sits in a sentence. This is inconvenient and leads to all kinds of imprecision, but so it is. (When you shout, "Hey you," you might find yourself clarifying whether you mean one "you" or a bunch of "yous." It's sad, really.)
Old timey English users had a way to differentiate between a "you" who is the subject of a sentence, and "you" who is the object. This is where "thou" and "thee" come in.
If the "you" starring in the sentence is the one doing the action, i.e. is the subject of the sentence, the word you use is "thou."
Here are some examples:
Wilt thou ever be a foul-mouthed and calumnious knave? (All's Well That Ends Well, Act 1 Scene 3)
Thou losest thy old smell. (As You Like It, Act 1 Scene 2)
Hast thou slain Tybalt? wilt thou slay thyself? (Romeo and Juliet, Act 3 Scene 3)
"Thee" is what you use when the "you" is the object of the sentence, the one unto whom something is being done. "Thou" is the action person; "thee" is on the receiving end. Examples:
I would not be thy executioner;
I fly thee, for I would not injure thee. (As You Like It, Act 3 Scene 5)
But he that temper'd thee bade thee stand up,
Gave thee no instance why thou shouldst do treason,
Unless to dub thee with the name of traitor. (Henry V, Act 2 Scene 2)
There are exceptions to this that you will find on occasion, where a writer will use "thee" instead of "thou." It is more rare to see "thou" used as a substitute for "thee." However, it's important if you're using old timey English to know that "thou" and "thee" were not interchangeable. If you indiscriminately use "thee" all over the place, as seems to be the favourite choice of modern writers, you will be sending up a burning flag that says "I don't really know what I'm doing." That is something thou shouldst not do, no matter how much it tempts thee.