Maybe just replace them with something random?
UPDATE [YourTable]
SET [Email] = REPLACE(CONVERT(varchar(36), NEWID()), '-', '') + '@fake.com'
That will give you something like:
3931BB9646304A949D9F4EBA81E9E798@fake.com
Or if your table has an identity column, plug that inherently unique value into a new fake e-mail address:
UPDATE [YourTable]
SET [Email] = 'email' + CONVERT(varchar(20), [Id]) + '@fake.com'
Or disable the constraint (without deleting it) before clearing out the column:
ALTER INDEX [YourUniqueIndexName] ON [YourTable] DISABLE;
Although that last one might alter the behavior of any application(s) writing to that table.