What are the reasons for autopilot restrictions on instrument approaches?
It's the fact the autopilot works more accurately than a pilot which is actually the cause of the restriction.

The decision to restrict the use of the autopilot usually comes from the certification agency after the ILS inspection flight detected glideslope erratic variations or reversals. The false signals are likely due to interference from the environment or the nearby activities with the runway equipment.

Flight inspection volume

The ILS volume surveyed by certification agencies using ICAO recommendations is located below a 5.25° angle. It means G/S indications are checked and expected to be correct only below this angle.


Source

This is why G/S are usually intercepted from below unless specific procedures are in place and extra care is given to validate G/S indications.

Anomalies in flight inspection volume

However there can be false and reversed guidance areas within this volume. They are discovered when the ILS is commissioned and flight tested by the certification agency. Example of glideslope reversal during an inspection on the nominal slope:


(Source, page 15-97)

In these areas, the ILS signal may vary quickly and if the autopilot were active it could react to the variations, while a pilot will likely not notice the interference, or will disregard it unless it follows a trend for a few seconds. A pilot with a Vertical Situation Display can confirm the ILS error visually.

A reversal, which is a guidance given in the wrong direction, can lead to a serious incident or an accident, due to autopilot quick reaction time. When such reversals are discovered during the inspection flight, autopilot approaches are not authorized, or forbidden during the final segment where the consequences can be the worst.

Consequences of a reversal during autopilot approach

Using the erroneous signal from the ILS may divert the aircraft from the approach path. Being below the glideslope close to the threshold is indeed not desirable, especially for a jet aircraft for which the thrust increase required to recover the lost altitude cannot be obtained instantaneously.

Suddenly jumping higher is also dangerous. It leads to a loss of airspeed, and possibly to an unrecoverable stall. Below is an example of autopilot reaction to a reversal. While it didn't happen in the certified volume of the ILS, the principle is exactly the one found in any reversal.

Some explanation about the context it happened

It was believed for a long time ILS radio signals for the glideslope contain false guidance at angles multiple of the nominal approach slope (e.g. 6° and 9° for a 3° G/S), but it was also taken for certain these false glideslopes were not reversals and would anyway guide the aircraft to the threshold, at a different descent rate (which would be enough to warn the crew about them following a wrong glide angle).

However G/S reversals, especially after 9° has been observed later (2011-2013), for some types of ILS antennas, without a full understanding of the reasons. Among these cases (source ICAO):


Schiphol Airport, The Netherlands, 2011, KLM, Embraer E190
Murcia Airport, Spain, 2011, Ryanair, Boeing 737-800
Charles de Gaulle Airport, France, 2012, Air France, A340 (BEA report)
Treviso Airport, Italy, 2013, Ryanair, Boeing 737-800
Eindhoven Airport, Netherlands, 2013, Ryanair, Boeing 737-800.


Eindhoven serious incident, May 2013

A B738 on autopilot was flying a non stabilized approach where the G/S had been captured from above. When reaching the 9° glideslope area, at 0.85 NM from the threshold, the aircraft suddenly pitched up to 24° and climbed from 400 to 1,000 ft. Airspeed plummeted to 100 kt, the stick shaker activated:


Source. Original source: Dutch Safety Board report

From the investigation done by the Dutch agency, it appears the actual guidance provided around 6° and 9° is completely reversed:


Source

As a partial mitigation measure, Boeing changed the Flight Control Computer software to limit the pitch up rate in G/S mode. The BEA also recommended manufacturers would prevent the capture of the G/S by an aircraft not on the published glidepath.