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They did for my Model X purchase. I provided a 10-day payoff letter a day or two before delivery, they applied the equity towards my purchase and had the lien paid off within the 10 days.
Tesla just takes bids from wholesalers for your car though, so a private sale might be better. Depending upon the value of your trade though, decreasing the tax burden of the Tesla purchase can offset the loss from not doing a private sale.