Why would there be flight restrictions when the barometric pressure exceeds 31 inHg?
It is because standard altimeters are not adjustable beyond this range. The AIM 7-2-2 contains information on how to handle it:


  
  When the barometric pressure exceeds 31.00 inches Hg. The following procedures will be placed in effect by NOTAM defining the geographic area affected:
  
  
  (a) For all aircraft. Set 31.00 inches for en route operations below 18,000 feet MSL. Maintain this setting until beyond the affected area or until reaching final approach segment. At the beginning of the final approach segment, the current altimeter setting will be set, if possible. If not possible, 31.00 inches will remain set throughout the approach. Aircraft on departure or missed approach will set 31.00 inches prior to reaching any mandatory/crossing altitude or 1,500 feet AGL, whichever is lower. (Air traffic control will issue actual altimeter settings and advise pilots to set 31.00 inches in their altimeters for en route operations below 18,000 feet MSL in affected areas.)
  
  (b) During preflight, barometric altimeters shall be checked for normal operation to the extent possible.
  
  (c) For aircraft with the capability of setting the current altimeter setting and operating into airports with the capability of measuring the current altimeter setting, no additional restrictions apply.
  
  (d) For aircraft operating VFR, there are no additional restrictions, however, extra diligence in flight planning and in operating in these conditions is essential.
  
  (e) Airports unable to accurately measure barometric pressures above 31.00 inches of Hg. will report the barometric pressure as "missing" or "in excess of 31.00 inches of Hg." Flight operations to and from those airports are restricted to VFR weather conditions.
  
  (f) For aircraft operating IFR and unable to set the current altimeter setting, the following restrictions apply:
  
  &nbsp;&nbsp;&nbsp;(1) To determine the suitability of departure alternate airports, destination airports, and destination alternate airports, increase ceiling requirements by 100 feet and visibility requirements by 1/4 statute mile for each 1/10 of an inch of Hg., or any portion thereof, over 31.00 inches. These adjusted values are then applied in accordance with the requirements of the applicable operating regulations and operations specifications.
  
  EXAMPLE-
  Destination altimeter is 31.28 inches, ILS DH 250 feet (200-1/2). When flight planning, add 300-3/4 to the weather requirements which would become 500-11/4.
  
  &nbsp;&nbsp;&nbsp;(2) On approach, 31.00 inches will remain set. Decision height (DH) or minimum descent altitude shall be deemed to have been reached when the published altitude is displayed on the altimeter.
  
  NOTE-
  Although visibility is normally the limiting factor on an approach, pilots should be aware that when reaching DH the aircraft will be higher than indicated. Using the example above the aircraft would be approximately 300 feet higher.
  
  &nbsp;&nbsp;&nbsp;(3) These restrictions do not apply to authorized Category II and III ILS operations nor do they apply to certificate holders using approved QFE altimetry systems.
  
  (g) The FAA Regional Flight Standards Division Manager of the affected area is authorized to approve temporary waivers to permit emergency resupply or emergency medical service operation.
  
  b. At or above 18,000 feet MSL. To 29.92 inches of mercury (standard setting). The lowest usable flight level is determined by the atmospheric pressure in the area of operation as shown in TBL 7-2-1.


So because most altimeters do not have this kind of range, the altitude readout will be inaccurate. This is especially problematic for IFR operations where you do not have visual reference to the ground.