Here are some hints for ensuring that the continuous stories game runs well.

1. Use a timer.  The game's no fun if one player just says, "Okay... let's switch now..." if he or she's starting to get restless.  The timer will ensure that players end mid-sentence and concentrate more on the story and less on when enough time has passed.
2. Don't let the time limit be too long.  Not only will the writers get restless with an idea, but there will be fewer writer changes throughout the game, and there will be less random silliness.  But don't let the time limit be too short, either.  If all you can get out is a sentence, the story will change too frequently for it to be very funny.  One minute is just enough time to get a good few lines of story out; I wouldn't go any shorter than a minute, but I also wouldn't go much longer than a minute and a half, depending on how fast the writers can write.  If you have a timer that's easy to set and dings or buzzes, all the power to ya, but the story works just as fine with a one-minute sand timer; just make sure that you or others are checking it periodically to ensure that you're not all being allowed too much time.
3. Write legibly!  If people can't read your handwriting, then they'll have trouble continuing your story and reading it aloud at the end of the game.
4. Fold neatly.  Come on, don't be a pig and crumple up the story.  Neatly crease the page along the line of the ruled paper.  Oh, and use ruled paper.  It makes the game much cleaner.  And don't accordion the page—unfold previous folds before making your new fold or the folds will bunch together and make for paper that is hard to write on.
5. Don't leave people with only one word.  My mom once left me with only the word "next," which isn't much to work with.  If there's an orphaned word at the beginning of a line when time runs out, there are generally two things you can do.  You can either write two or three more words, or fold the story back another line.  Either way works good, it depends on how much material you want to give the next writer.
6. Don't peek.  You shouldn't look at what was written before the fold.  What, are you trying to ruin the game?  No sneaky reading the rest of the story; you'll have to wait until the end of the game.
7. Try to avoid excessive dialogue.  If time runs out when you're in the middle of a character's dialogue, the next writer may not know that a character's speaking and won't end the quotation marks, essentially making the whole thing one big piece of dialogue.  Well, actually, this isn't as bad as it may sound.
8. Don't be concerned about gender.  If the previous writer was writing about someone named Sally, but you didn't know that because it was folded away, don't be upset if you learned that the character you were calling "he" is female.  Other readers understand that you didn't know the character was female.  Along the same lines, sometimes a character's name can change too.  Just laugh about it.
9. Variety is good.  It's a good thing if writers have different interests.  The story could change from fantasy to science fiction to something cheerful and then into something morbid.  The changes between different writing styles and moods are one of the best parts of the story.
10. Don't write the same stuff over and over.  It gets boring if you write the same sentences every time.
11. Know when to stop.  I've ended the game in two different ways.  The way we used to end the story was after every player had written on each page.  So after your original page returns back to you, you could end the game then.  More recently, we've been ending the story when one player is getting close to the end of the page.  This should allow for a longer game.  One player says, "this story's getting near the end of the page" and we agree that it will be the last writing session.
12. Finish the story.  You don't want the story to end mid-sentence.  Wrap up the story to make it seem finished.
13. Read the story without mentioning line breaks or writer changes.  It's no fun if you mention where one writer's story ends and another's begins.  Everyone will know when the writer's changed because the story will usually totally shift gears.  Just read the story as it's written, and pause for laughter.  You'll probably have to pause anyway, because you'll be cracking up with laughter as well.