What, are the mongol hordes going to storm the doors as soon as the doormen walk off?
Realistically, the tenants want and expect their deliveries, which the doormen screen, enable, and hold for pickup where appropriate. Tenants would be wise to postpone deliveries, if possible, during the potential strike, because volunteers will probably screw it up.
In my own building, which has no doorman, not only has there been no strike-related updates for a month, most tenants have, like myself, decided that if the strike lasts for less then a week, none of us have to think about our garbage.
This really is a strike aimed at the more affluent. For the few couple of days, the co-op partners filling in will get their props. Old habits will re-establish themselves rather quickly, I'd wager.
[Update: strike averted!]