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I am buying a new car. I was planning on paying cash. There are various dealer incentives if I finance - worth maybe $750-1000. If I take out a loan and pay it off within a month or two, will this hurt my credit score?
The only impacts to your credit score would be from a new account ding and a possible decrease in your Average Age of Accounts. Both of these are relatively minor. While it is impossible to predict what the exact drop could be even with the details of your credit history you are most likely looking at 10-20 points immediately and gaining these back over the next months. Especially if the account is paid in full in a short term.
On the other hand if you have no active installment loan history it could actually add points. If you wanted to you could simply pay back almost all of the loan in a short basis (like 90%+) and let the loan ride out for a while before you pay off the remainder. At a decent interest rate you would still pay way less interest than the incentive cash.