Just found the solution to this problem (at least when it is caused by typing from an iPhone):
Go into your iPhone Settings>General>Keyboard and turn "Smart Punctuation" OFF. Smart punctuation substitutes alternate characters for some of the standard set - including apostrophes, and quotation marks (for example, substituting a curly apostrophe or curly quotes for a straight ones). Many web sites don't understand these alternate characters, or may not have them in their chosen default font.
Typed on my iPhone with Smart Punctuation ON:
I don稚 like 鉄mart Punctuation? It痴 always messing things up.
Typed with Smart Punctuation OFF:
I don't like "Smart Punctuation". It's always messing things up.
If you look closely when typing from an iPhone, you'll notice that when Smart Punctuation is on, apostrophes and quotation marks appear curly when you post them. When Smart Punctuation is off, they are just little marks that go straight up and down. (You'll only see the curly quotes - and they will appear normal - as you are typing, once you hit post, it goes to the TBN site, which can't handle those characters, so it appears messed up upon later viewing.)
If you copy and paste from a program that also uses these characters, you'll also have the problem. If that application has the ability to turn off "Smart Quotes" or "Smart Punctuation", doing so will make your cut & pastes come out clean on this site.