My credit score dipped ~20-30 points when I finished paying off my student loans, but I think it bounced back within a few months. If it's installment (car/mortgage/student loan), it'll probably dip a bit. If it's consumer ($10k debt on a credit card), I think it would jump up quite a bit.
(The following is according to Mint)
The largest factor contributing to your credit score is your credit card usage - what's the ratio of your use to your total credit limit. This doesn't necessarily mean your balance by the end of your statement, by the way, a credit card can report your usage whenever they want. So if you have $10k on your credit card and a total limit of $20k, that's a 50% usage. Mint recommends keeping it under 20%. So if that $10k goes away and you're left with $0 (or, more realistically, >10%), your credit score shoots up.
Another factor is the number of accounts open in your name. If it's a car/student/mortgage loan that'll close when you finish payment, you'll lose an account in your name, which could decrease your score. This is a less important factor than usage, but does still count. (Apparently, 22+ accounts is excellent, 13-21 is good, 6-12 is not bad, and 0-5 is poor.)
One more factor is the age of credit. This is the average age of all of your accounts. So if you've had this loan for a long time and it closes, your average age will decrease, which will knock your score a bit. However, if it's relatively recent, getting rid of it could increase your average age, increasing it.