What are the SOPs following an overweight landing?
The answer very much depends on the operator and type of aircraft (or rather its manufacturer).

First of all, overweight landing would normally be performed for the following scenarios:


Medical emergency on board that requires prompt medical attention
Any fire/smoke that cannot be extinguished/ventilated
QRH dictating "Land at the nearest suitable airport"
Any situation where safety might become further degraded by prolonged flight


It's hard to list all the situations here, and most of the time the Captain will have to make a decision whether to do an overweight landing or not.

Having made the decision to perform an overweight landing, the crew needs to take into consideration the following items:


Runway length available - landing at significantly higher weight than MLW increases landing distance substantially
Missed approach climb gradient - this might be a problem at airports with high terrain nearby
Reduced landing flap - normal landing flap speeds at higher than MLW might be too close to Vfe (maximum speed for the selected flap setting)
Required brake cooling - if heavy braking is required to stop within the required landing distance available, then it's best to vacate the runway and have fire services check the brakes and tires. Fuse plugs might have melted and tires might have deflated. Subject to all clear from the fire services, aircraft might taxi to the stand, have the chocks set and wait there with brake released for a while.
Overweight autoland is normally not recommended, as the autopilots are not certified for this.


After an overweight landing, an inspection is required by engineers to make sure there are is no structural damage on the aircraft, particularly its landing gear and the skin. According to my knowledge, the overweight landing inspection is just a bit more thorough walkaround by the engineer, provided the overweight landing wasn't a hard landing. If it was a hard and and overweight landing combined, the aircraft is likely grounded for a while, as more detailed inspection is required.

All Part 25 (FAR 25, CS 25) aircraft are certified to be able to land with sink rate of 600 ft/min (10 ft/s) at MLW and 360 ft/min (6ft/s) at MTOW, so unless you do a particularly poor landing when overweight, there should be no structural damage, even when landing straight away after a takeoff at MTOW.