$ awk
  '$0 ~ /^[0-9]{4}-[0-9]{2}-[0-9]{2} [0-2][0-9]:[0-5][0-9]:[0-5][0-9]/
      {
        if ($1" "$2 >= "2014-04-07 23:00")     p=1;
        if ($1" "$2 >= "2014-04-08 02:00")  p=0;
      }
    p { print $0 }' log