When you sell your home for less than you owe your mortgage company
and they won't allow you to redo the loan and pay off the rest, how is
your credit affected?  No late or missed payments are involved either.

Bellagio,

Why would you sell for less than you owe? If you haven't missed
payments et, al. You are throwing money away, or giving it to someone
else.

Sell you house for what you. If you are short, you will still have to
make up the difference to pay the loan off.

e.g Owe 100K
Sell for 80K

You will still have to satisfy the 20K difference.

Clint, it's sometimes legit to sell a home for less than is owed on it
(although that usually means a bad or uninformed decision was made
prior to that).
Bellagio, from your question I'm guessing that the mortgage company
wants you to keep making the regular monthly payments on the remaining
balance until the house is fully paid off.  This will not look bad on
your credit rating.

Your FICO Score (your credit rating) is based on:
35% Payment History
30% Amounts Owed
15% Length of Credit History
10% New Credit (# of recently opened accounts and inquiries)
10% Types of Credit Used
http://www.myfico.com/CreditEducation/WhatsInYourScore.aspx?fire=5

Keeping a balance from your old home will only effect your Amounts
Owed (however you owed the amount before anyways so it's not really a
negative impact).  And home loans are looked upon positively as far as
"Types of Credit" are concerned, so keeping the loan could actually
boost your score slightly.

I hope your interest rate is fairly low since you're stuck with the
loan.  And unfortunately the interest is no longer tax deductible
since you don't live in the house.

according to our mortgage company they won't let you pay it back.  You
do what is called a short sell.  We have to do it because we are
carrying two payments now and the market where we need to sell is not
good as it is in other places.  We are talking about suburban Dallas,
TX, where you cab buy a 3000 sq ft house for 275,000!