Use null value to make the Date field null.
Sample Code:
Account objAccount = [
SELECT Id FROM Account WHERE Name = 'InfallibleTechie' LIMIT 1
];
Asset objAsset = new Asset(
Name = 'Testing',
AccountId = objAccount.Id,
InstallDate = Date.today()
);
insert objAsset;
System.debug( 'Asset Install Date is ' + objAsset.InstallDate );
objAsset.InstallDate = null;
update objAsset;
objAsset = [ SELECT InstallDate FROM Asset WHERE Id =: objAsset.Id ];
System.debug( 'Asset Install Date is ' + objAsset.InstallDate );
Output: