One particularly pervasive example appears in the title of this article: the substitution of "ye" for "the". No speaker of early Modern English would do this, as it derives entirely from the fact that the Old English letter "thorn" (Þ) was used to render "th" in writing at the timenote and even then, it had already fallen so far out of favour that "the" was pretty much the only word it was used for, and in some Fonts (especially blackletter), it looked very much like a "y". Since the patent on printing presses was German, and England's earliest printers imported types from the Netherlands, the presses lacked the Anglo-Saxon letter, and "y" was substituted instead. Don't confuse this with the second person plural pronoun "ye," meaning "you," as in "Gather round, ye lads and lassies," which is a different word entirely and is pronounced as written.
Similarly, in the English-speaking world until the early 19th century, the letter "s", at the beginning or middle of a syllable, was written as with the long "s": ſ The character ß (Eszett or sharp "s") in modern German (pronounced and sometimes written "ss") originated as a ligature of ſz. In addition, the integral symbol and the IPA letter esh (representing the "sh" sound) were both derived from the italic version of ſ and look nearly identical note well, at least in typefaces that have proper italics and not just obliques.
Don't expect most people to correctly distinguish thou art/wast from thou beest/wert. The former pair are in the indicative mood, whereas the latter are in the subjunctive.
Quakers notoriously adopted use of "thee" as a pronoun as part of their tradition of "plainspeaking", in order to make the point that they eschewed all forms of flowery respectful formal speech, even ones that were centuries old and no one thought of as "respectful" anymore like addressing people as "you". The unkind stereotype, of course, is that since they started doing this in the 18th century long after "thou" had passed out of common use they did so incorrectly — "Quaker speech" stereotypically just uses "thee" all the time without regard for nominative or objective case.
"Whence", "thence" and "hence" mean "from where", "from there" and "from here" respectively. Therefore, saying "from whence" is redundant, although it's hard to argue that it's outright wrong since it does have precedent in the works of Shakespeare and the King James Bible.
Some writers seem to think that "thou" can be plural, possibly because they just think it's an older word for "you", and "you" can be plural.
Getting the cases wrong. "Thou" is the subjective case, and "thee" is the objective. "Thy" and "thine" are possessives. In particularly wretched examples, you may see "thou", "thee", "thy" and "thine" used interchangeably, which is pretty much the second person equivalent of using "I", "me", "my" and "mine" interchangeably.
Using "mine" and "thine" in front of consonants other than "h". They were used in front of vowels and "h", almost like "an" is used today.