The multimeter almost certainly hasn't 'failed', it sounds like it's working perfectly.
There are two types of mains appliances, Class I (earthed) and class II (not earthed), the meter will be class II, as an earthed multimeter would be a VERY serious safety hazard.
Because Class II devices aren't earthed they can gain a high static charge, to prevent this there are special safety components connecting from chassis to the incoming mains - these are usually a high value resistor and a capacitor in parallel, and are vitally important, and drain the static charge to ground (via the mains).
The result of this is that the chassis of the appliance floats at about half mains potential, as a VERY low leakage current, hence the small tingle you can feel.
Perfectly normal, nothing to worry about - and certainly something you shouldn't try and 'cure', as you would be making it dangerous.