The answer is YES, but not always!
Sometimes the first we know about a dampness problem is when your electrics trip.
The cause of dampness depends on how water has penetrated the wall – this could be rising damp, penetrating damp or downward dampness and usually occurs on outside walls, and down the side of chimney breasts (where there are often sockets in older properties).
You would need to seek professional help on how to resolve the dampness aspect.
Can damp affect electrics in walls?
In terms of your electrics, tell-tale signs include rusty screws (either side of the socket faceplate), and if the socket is a metal one, you might see green corrosion and the plating peeling off.
When the socket is removed, the galvanised backbox (that holds the socket in the wall) may be completely destroyed (depending on the age and degree of the damp problem).
Can damp cause electrics to trip?
Sockets: Now what can happen to your socket is that sufficient moisture getting into the socket could cause current to track between the live and neutral (causing a short circuit), or live to earth (causing an earth fault).
Cables: In terms of damage to cables, this is more unlikely as cables are hermetically sealed, unless they were damaged during installation or damaged due to later building work or DIY activity. If the sheath becomes damaged, water could get in, causing the protective device (fuse, breaker ort RCD) to trip.
Are damp conditions electricity hazards?
All circuits have some sort of protection. As a minimum, you will have a Fuse or Breaker that protects the circuit against short circuit faults.
But most properties nowadays have additional protection in the form of the RCD which gives protection against earth faults (which protect humans from electric shock).
To see if you have RCD protection, you can identify this by looking at your board:
How Do I Resolve This Electrical Issue?
To start with, you really need to get a builder or specialist to resolve the water or damp ingress.
Any damage to sockets or cables would then need to be resolved by an experienced electrician.
There are a number of solutions that an electrician could offer depending on the situation.
If you’d like to discuss the solution to such a problem, please get in touch »
WEEE is Waste Electrical and Electronic Equipment. It is end-of-life electrical and electronic equipment and covers virtually everything with a plug or battery. 
Fuses (the old plug in style) or MCBs (newer switch style circuit breaker (herein called breakers) are designed to protect your property’s internal wiring, while RCDs protect people from electric shock. 
In the UK a substantial number of people die and others are injured in domestic fires every year, many from smoke inhalation. 
Emergency lighting is lighting for an emergency situation when the main power supply is cut and normal illumination fails due to a fire or a power cut 
UK Power Networks own and maintain the electricity lines and cables that power over eight million homes and businesses in London, the East and South East of England.