Will an insect getting underneath the keyboard cause problems?

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Yes, insects can cause problems. It also indicates possibly that food was around the keyboard one too many times. My college 16 years ago was adament not to allow food or drink in the computer room, because of having to constantly replace keyboards. You can imagine how much more sensitive notebooks might be, especially if they have backlighting.

Insects inside computers and other electronics are a well-known cause of trouble. It's unlikely that a single ant would be likely to do substantial damage, but ants rarely travel far from other ants, and they leave pheromone trails for others to follow. The warmth of the electronics attracts them, and the electrical fields may add to the attraction. Ants can cause corrosion and short circuits, impede cooling, and Apple has even been known to refuse to work on machines that are heavily infested with bugs.

No, the computer isn't designed to keep insects out. It can't be: it has to ventilate itself. Even with a keyboard protector on it, all the port openings and the intake and outflow vents along its back edge (mostly hidden by the display hinge) would be wide open, offering unobstructed routes into the heart of the machine. A keyboard protector will, however, keep crumbs and dust out of the keyboard, and though this is my sixth Apple notebook and I've never used a keyboard protector before, I put one on this MBP the moment I took it out of the box and haven't typed a keystroke without it in 23 months. There isn't so much as a single fingerprint on my keyboard. I decided to use a protector on this machine because it's so much harder to clean underneath the keycaps with the aluminum grid covering the spaces between them.