well, you asked "What parts of the MacBook Air are really sensitive"
not "HOW is the easy way to damage one"
The easy way to damage one badly is from a spill.
the SENSITIVE parts, as you asked are the LCD display. They are by nature rather fragile creatures to be certain.
Cleaning Your MacBook
Use a very lightly damp, soft (if you can squeeze the cloth very tightly and even a drop comes out, its still too damp), lint-free cloth to clean the computer’s exterior. Avoid getting moisture in any openings and be very careful not to get the damp cloth around the trackpad edges or around the inside edge between the monitor and its bezel. Do not spray any type of liquid directly on the computer. Do not use sprays, solvents, or abrasives; do not attempt to use any solvents on the LCD display either directly (especially!) or indirectly. There is never any call to use any type of lens cleaning solvents on the screen of your LCD display. Again, do not spray any liquid directly on the screen. Never use Kleenex, or paper towels, these are abrasive. Soft cotton cloth only or a microfiber lens cleaning cloth.
No rubbing alcoholic, no glass cleaners of any variety, no acetone, no lens cleaning sprays, no AR coating sprays, no sprays sold as "for your LCD TV", no sprays sold as "for your notebook display". Water only, and never sprayed on, and never dripping wet even slightly on the cloth.
Use a soft paintbrush, lipstick lens-cleaning brush, or makeup brush for cleaning
One very useful tool for cleaning dust from the LCD screen, the hinge area of your notebook and around the keyboard, is a very soft inexpensive small paintbrush or a makeup brush. The less you wipe your LCD screen with anything like a cloth, the better. This small brush is very handy, quick, and ideal for removing dust from around and on your notebook. Lipstick lens cleaning brushes are sold online, and used in the photography business for camera lenses and are ideal perfect portable cleaning brushes for your macbook.
Dont press on your LCD screen or spray it
Do not attempt to use any solvents on the LCD display either directly (especially!) or indirectly. There is never any call to use any type of lens cleaning solvents on the screen of your LCD display. Do not spray liquid directly on the screen. Avoid any touching or pressure upon your LCD screen.
Also NEVER pack your Air in a backpack where books can press against the LCD assy.
Dont 'backpack' your Macbook Air without having special consideration
Many college students or people in general are fond of using non-idealized backpacks (without a padded laptop compartment) along with their books. Sandwiching books around your notebook etc. is an extremely bad idea and can lead to your LCD screen being destroyed when heavy textbooks are pressing against same, or when dropping your backpack, these books slam into your Macbook causing great potential for damage.
The other "sensitive" part isnt sensitive anymore ,the USB ports are upgraded in 2013 and are hard to cause tab damage on.
The other "sensitive" part would be the fan and dust bunnies.
Dust and fuzz residue caked onto the internal fan

Dust bunnies and Fluffy’s fur
Your Macbook’s cooling system, including fan and rear vents pull cool air in from the back and (in the case of the Macbook pro) also from the left ports bringing this air around the processor, GPU, and heat sync and blow it out the back, as such your Macbook acts like a little vacuum cleaner of the air around you. If you use your notebook in a dusty area or around a lot of pet fur, this can quickly, even under seemingly ideal clean rooms, accumulate inside your notebook especially on the fan blades reducing its capacity for cooling and in a worse case scenario seize your fan due to caked on dust on the fan blades. Avoid very dusty areas or places where, over the long term you know fuzz, fur, or even smoke can enter into your Macbook.