Can you legally remove the engine of your financed car before it is repossessed?

No, in most places it is specifically listed as a crime and in others would be treated as conversion, theft or vandalism, depending on the laws of the jurisdiction, the value and the agreement signed.

In Texas, as one example, the law states "A person who has signed a security agreement creating a security interest in property or a mortgage or deed of trust creating a lien on property commits an offense if, with intent to hinder enforcement of that interest or lien, he destroys, removes, conceals, encumbers, or otherwise harms or reduces the value of the property."

Removing the engine would definitely reduce the value of the property.