Allowing buyers to move in early seems like a nice gesture that maybe helps them out of a bind, but quite honestly there are too many potentially negative consequences for you to justify even considering it.  If the bad stuff happens, it could cost you a ton of time, money, and even the transaction itself.
 
Real estate deals disintegrate in the 11th hour everyday due to all sorts of factors.  If for example, the buyer’s financing falls apart the day before closing, how are you going to get the buyers out of your house?  At that point you would be required to go through the eviction process because they would now be considered tenants.  Even if you manage to get them out, what if they damage the house?  How will you make them pay for it?  Are you prepared to file a lawsuit?
 
Even if your deal stays together, you still are at risk.  If the buyer’s dog bites the paperboy while you still own the home, as the landlord you can be held financially liable for Fido’s actions.  If any issue or dispute arises in your transaction, the buyer has additional leverage in negotiations because they are in possession of the property.  If things get acrimonious, do you really want people upset with you living unsupervised in your house?
 
There is just not an upside worth the risk of allowing an early move-in.  In fact, the one good thing that could come from this decision—the closing happening as planned without incident—will almost surely happen just as readily without the early move-in.