What should I take care of while using elastomeric connectors?
General electronics handling precautions should be observed:


Once exposed the elastomeric structure is a good capacitor and can accumulate static charge. Ground yourself prior to working with the parts to avoid an ESD event which could damage the onboard IC's inputs. Maintain grounding. Avoid working in a carpeted area. Etc...
Wash your hands with soap (and rinse thoroughly) and then isopropyl alcohol to remove contaminants. Even so, avoid contact with the connector's and pcb's electrical contact area.
An oxidation layer will form over time and alter the contact impedance. Therefore, you should minimize the exposure time and work in as dry a room as possible. That's a practical recommendation that deserves some qualification:


As @embedded.kyle very keenly and correctly notes (Thanks Kyle!): "There are certain ranges of humidity that strike a good compromise between the increased risk of oxidation and the increased risk of static buildup. Humidity below 40% starts to increase the risk of static generation. NASA-STD-8739.7 recommends between 40% and 60% humidity. ANSI/ESD S20.20-1999 and MIL-STD-1695 are a bit more relaxed at 30% to 70%."
If you are serious about long-term reliability, you should adhere to these climate control standards. For a hobbyist-grade project, I recommend working in "as dry a room possible" given that, in the US at least, natural weather patterns are more likely to leave you too humid than too dry and static control techniques can mitigate the risk, whereas there is little you can do on the other end of the spectrum.



Tips on alignment:


Elastomeric's are typically classified into two categories, those with contacts substantially smaller than the PCB pads (very tolerant of alignment error), and those approximately the same size (alignment sensitive). Unfortunately, for LCD applications, the latter type is the most common since the manufacturer needs the connector density to increase the communication bandwidth to the display.
Alignment can be extremely challenging. Rely on the original alignment mechanism if possible. If you can avoid breaking the original contact seal, all the better (then you won't have to re-align it). Some devices have clips and other apparatus to assist alignment (although this is becoming increasingly rare in my experience).
If you must reattach, fashioning a makeshift pick-and-place machine out of a mill (CNC or otherwise) and an eye-dropper or baby bulb (hand operated vacuum pump) that you tape to the chuck (or head) can give you the precision you need to re-align. Use of a dissecting microscope or loops is also helpful as it improves your mechanical acuity.


But above all, may you have GOOD LUCK! =)