There is a wealth of information on ESL's at diyaudio.com under the Planars & Exotics forum. The technical guru's there are extremely helpful and generous with their knowledge for both the novice and the more experienced enthusiasts. They have helped me with mine on several issues I've had. The Martin Logan owners forum should also have plenty of info.
Now I am not technical, but I will give you my understanding of what can go wrong with ESL speakers:
Because a normal amplifier does not produce enough volts to properly drive an ESL, a step up transformer is used for this purpose... anywhere from 50 to 150:1 ratio. If over driven, they can fail causing low output and distortion...kind of a crackling noise. This happened to me one night on my left speaker while listening to some very loud and dynamic music. However, this was pilot error on my part since I wired the transformer for maximum output vs a lower safer output(I'm still mad at myself for this!). I don't think Martin Logan's have this problem?
A bias power supply charges the diaphragm to a high voltage(2000-4000 volts!) so it can interact with the stators-the curved perforated metal on your speakers. If the power supply fails, there will be less sound or hardly any sound at all.
The thin mylar diaphragm vibrates by being attracted to the + or - stator, producing music. Over time, the conductive coating may lose its effectiveness or actually separate from the diaphragm. Dirt, dust, and bugs can find their way between the diaphragm and the stators. Bugs at nighttime can be quite entertaining with a light show as they are electrocuted...but this may cause a burnt hole in the diaphragm. The ML's should be well insulated to prevent this?
The stators are usually trouble free, unless they somehow lose their insulation or bent too close to the diaphragm.
These are items that may go wrong but most likely will not. Hope this helps a bit. Look into the forums for the real experts, though!
Bond-san