The housing firm will charge more for apartments and pay less for its inputs, so the tax will actually be paid by consumers and input suppliers.
A unit tax increases the cost of production, so we need a higher price to get firms to produce any given quantity.
The shift of the supply curve increases the equilibrium price of apartments.
The apartment tax affect the people who supply inputs such as land and labor to the housing industry.
The tax decreases the output of the industry, so the industry needs smaller quantities of inputs used to produce apartments.
The resulting excess supply of inputs will decrease land and labor prices, decreasing the cost of producing apartments.
If the demand for tax goods is inelastic we need a large price hike to eliminate the excess demand caused by the tax.
Deadweight loss from taxation is the difference between the total burden of a tax amount of revenue collected by the government.
Excess burden of a tax is another name for deadweight loss.