Can I hook up a electric 220 dryer,using a pig tail to a 110 outlet without harming the dryer or my apartment

A simple answer. NO

Chuck

Hhow do you plan to get 240 volts?

No, you cannot use a 120Vac to feed a 240Vac clothes dryer. As a tenant, you are not allowed to do any electrical work at all. A licensed electrician must do the work. The Landlord is the only one that can authorize such work.

Physically it cannot be done. The dryer requires a higher amperage then the single outlet can deliver without causing a fire and the voltage has to be supplied by both phase bars in the service panel. A 120 Vac receptacle is only using one of those phase bars.

You would have to het your landlord to agree to pull a 240Vac - 30 Amp circuit from your apartment panelboard to where the appliance will be. Also, this will require that a dual breaker be installed which means that you would need two open slots for the new breaker.