What problems could occur if I plugged a 3-prong plug into a receptacle labeled "No Equipment Ground"?
Any equipment that uses a two prong plug will be as safe being plugged into this setup as they would be if they were plugged into a properly grounded three prong outlet. Two prong plugs (obviously) do not use the ground pin. As for three prong plugs, this depends on what is being plugged in. If the item is a laptop power supply that is sealed in a plastic shell, the ground pin is pretty much ISO-proforma [meaning, that is the standard cord- whether the equipment needs a ground or not.] I would be hesitant to plug in something with a metal shell, like a full size desk top tower, because the power supply is enclosed in a metal shell and attached directly to the frame of the case, this could present a shock hazard if the power supply were to malfunction. 

A GFCI is a good start. If they are functioning properly, they will guard against lethal shocks, but you may still get a tingle if the electric is not quite right.

A possible suggestion: Use a 2->3 pin converter and attach a ground wire to the grounding tab. Attach the other end of the wire to a radiator if you have steam heat or possibly a metal water pipe. (no gas pipes.) This will get you a ground and you can use those items safely.