Why are there speed restrictions applied after deicing an aircraft?
Some deicing fluids can leave a gel-like residue on the airplane. Several incidents occurred with 737s where the residue accumulated in the elevator balance bay, causing severe oscillation/vibration in flight above 270 KIAS. These incidents resulted in the issuance of Airworthiness Directive (AD) 2002-08-20, which required adding the 270 KIAS limitation you discovered into the 737's Airplane Flight Manual (AFM). The required "maintenance procedures" consist of cleaning off the deicing fluid residue. The AD also requires a one-time cleaning and sealing of the elevator balance bay.


  The FAA has received reports of numerous incidents of
  severe airframe vibration, or limit cycle oscillation (LCO), in flight after the horizontal stabilizer had
  been deiced/anti-iced on the ground. The reported incidents occurred on Boeing Model 737-600, -700, and -800 series airplanes. The empennage structure on these, as well as Model 737-700C series
  airplanes, is identical; therefore, all of these airplanes are subject to the identified unsafe condition.
  These events have been attributed to an accumulation of deicing/anti-icing fluid or other residue in
  the elevator balance panel cavities and on the external surfaces of the elevator tab.




A Boeing article on the subject gives some more information and also has some photos showing the residue. Here's what a gummed-up 737 elevator balance panel looks like:



From the article:


  Airplane deicing and anti-icing fluids can leave residue in critical areas in the wings and stabilizers. This residue can rehydrate and expand into a gel-like material that can freeze during flight and cause restrictions in the flight control systems. Therefore, attention to this residue should be part of a regularly scheduled inspection and cleaning process.


...


  Although Type III fluids have not been specifically linked to any events involving flight controls, the composition of these fluids makes them equally susceptible to residue problems as the Type II and Type IV fluids.