It is definitely ungrammatical to say "*The management has never and will never closed the door to negotiations." The auxiliary "will" cannot be followed by a past participle; it has to be followed by an infinitive.
It is grammatical to use both forms explicitly, as in "has never closed and will never close."
A Google Books search reveals that people do also use forms with ellipsis such as "has never and will never close." Some might hesitate to use this structure since the explicit verb "close" doesn't have the same form as the implied verb (closed), but such mismatches can also bee seen in other elliptical sentences, such as "The management has never closed the door to negotiations, and it never will (close them)."