I have an ungrounded circuit in my house with three-wire grounded receptacles. If I install a GFCI, will the downstream receptacles be protected even if the GFCI is ungrounded?
Absolutely, as long as you place the load wires going to the downstream receptacles onto the LOAD terminals of the GFCI receptacle.
A GFI does NOT need an equipment grounding conductor to function properly. In fact, using a GFI at the beginning of the circuit is what will allow you to legally and safely use 3-prong receptacles downstream. Thing is, this does not create an equipment ground for equipment protection for things such as surge suppressors.
By code you are supposed to label any non-grounding downstream receptacles "No equipment ground present".