If I use the "Rent Furnished" option, will the furniture still be lost when the family moves out of the apartment? If so, is there a way to prevent it?

The way apartments work is, when you build it, each unit becomes a template of itself. When rented, a brand new unit is generated from that template. When a family moves out, this unit is destroyed and when it is next visited or rented, the template will be used to make a new copy.

So the furniture that is there when you rent it furnished will be there, in the original template, when the sim moves out, and when a new sim moves in. But any changes that were made by the original renter will be lost.

To add on what Peni said, no, there is no way to prevent it. In residential lots we can use the stay things shrub to prevent things from being cleared when they move out, but like Peni said, when you move out of an apartment the lot the Sim was actually living on (from the games point of view) is destroyed. It'd be like trying to keep the furniture on a residential lot when you delete it. Can't, because it's no longer there.