We still don't have an answer as to what happened to Lanfear, just clues which you can decide to believe or not.

According to Moiraine, the Finns fed off of her directly. This is very similar to when Mat was before them and they enjoyed the savour. It does seem that channelers are even sweeter for them, and that they can feed directly off of the channeler's ability. Moiraine has been slowly drained and is now very weak in the Power, which proves that they can drain the channeler's ability to permanently. This is a remarkable ability if you think about it.. The Forsaken can't accept that Cyndane is Lanfear because she is weaker in the Power, implying that they are unfamiliar to any method whereby channelers are weakened permanently. We know that Imperfect Healing of stilling has this effect, and for a time this was a theory surrounding Cyndane's weakness. Other theories have surrounded complex shielding, especially Moridin perhaps using the TP on her.

Moiraine says that they drained Lanfear too quickly and killed her, but she thinks that this is a lie made to frighten her.

I personally believe it is the truth.

Moiraine's statement implies that she never saw Lanfear after they fell through the doorway, because she relies on news by the Finns to remark on her. It is thus most likely that the two of them lost consciousness after they fell through the door. This is not really unexpected considering so much of the Power was being used and it involved a ter'angreal. Many readers thought that both of the channelers were stilled or killed at that time. It also explains how they captured them. Rand proved early on that even with a fire sword they were no match for him, and Moiraine and Lanfear are no easy meat.

For some reason they drained Lanfear too quickly. It could be because they are inexperienced regarding the process and Lanfear was their first victim or perhaps such a rare treat made them ravenous and they piled in. It is also possible that Lanfears great strength the Power was the factor that drove them to overdo it... A bit like landing the fattest turkey in the land. It is unlikely that they killed Lanfear deliberately either because she was part of the Shadow, which they hate, or because she cursed or mocked them... Moiraine seemed rather comatose in her draining fog.

Lanfear actually got a good deal. If she'd been drained slowly like Moiraine I suspect she would still be a captive, and about as weak as Moiraine is. I find it unlikely Moridin or SH would rescue her unless they saw a direct need as they did with Semirhage. Her almost instant death lost her some strength but prevented further loss in the end.

All in all, I think we have been given the answer as to how events played out regarding Moiraine and Lanfear. Moiraine thinks its a lie but it probably isnt.