Why does in the hydrolysis of acetylsalicylic acid with NaOH the solution has to be simmered gently?
What could happen if I would strongly boil it


Mechanistically, nothing different will occur.  The same reaction that occurs at lower temperature will still occur at higher temperature.  The only difference will be the rate of the reaction; the reaction will go faster as you increase the reaction temperature.  In a classroom setting the procedure might involve simmering rather than boiling because


The reaction is usually followed by making uv-vis absorbance measurements every 15, or so, minutes.  Depending upon the reaction temperature and the concentration of base, the reaction might proceed too fast at boiling to allow for convenient measurements (e.g. the reaction might be complete in 10 minutes).
Safety reasons, if something goes wrong (the flask cracks) a warm basic solution won't burn the student as bad as a boiling basic solution would.