Yes, they do, and it can indeed pose significant problems for aircraft.
Asphalt runways are susceptible to ruts (because asphalt isn't really solid - it's solid aggregate suspended in an extremely viscous liquid). The friction (heat) from landings partially melts the asphalt, and the weight of the aircraft slowly pushes it outward away from the landing gear, creating ruts. (There are also other ways you can cause ruts on asphalt runways, such as landing overweight aircraft on them on extremely hot days).
Concrete runways are theoretically less susceptible to rutting, but have their own maintenance challenges.
Runways can also develop potholes, cracks, and spalling - just like a road surface.
Because a pothole can cause loss of control on landing (if the landing gear falls into it and blows a tire or yanks the plane to one side), or serious damage to aircraft engines (if some of the material ejected from the pothole gets pulled into a jet engine's intake or kicked up by a propeller) airports take a lot of effort to prevent (and where necessary, repair) this kind of damage, and routinely inspect their runways.
Airports also do a lot of preventative maintenance on their runways, sealing open cracks and joints in the pavement material to prevent water from infiltrating under the runway and causing potholes (see the image below - click to enlarge).
The FAA has a lot of information on runway pavement design, which I will affectionately call "More than anyone ever wanted to know about runway design, and never asked!"