Will alcohol or soap damage plastic or rubber?
Will soap damage plastic or rubber?


Probably not. Most plastics and rubbers are resistant to aqueous solutions and ionic compounds (including most surfactants and soaps).


  Will alcohol damage plastic or rubber?


Maybe. There are two problems, in that neither alcohol nor plastic refer to specific compounds, but are rather generic terms encompassing hundreds of compounds each. The number of possible combinations is endless. As Leonardo suggests, the best approach is either to look in the manual or experiment in a small inconspicuous part of the product.

However, assuming that by alcohol you either mean ethanol, which is the drinking kind of alcohol, or isopropanol, which is the alcohol in many cleaning products, then I can give you some advice based on common plastic types. Both are similar enough in their solvent properties that we can generalize.  

Alcohol will damage some plastics, but not all. I will list my answers by the resin code, or "recycling symbol", found on most plastic items:


Poly(ethylene terephthalate), PET or PETE - PET is not very soluble in ethanol or isopropanol, but prolonged exposure may cause crazing or stiffening due to the dissolution of plasticizers.
High-Density polyethylene, HDPE - HDPE is resistant to most things.
Poly(vinyl chloride), PVC - PVC is not very soluble in ethanol or isopropanol, but prolonged exposure may cause crazing or stiffening.
Low-Density polyethylene, LDPE - LDPE is resistant to most things.
Polypropylene, PP - PP is resistant to most things.
Polystyrene PS - PS is not very soluble in ethanol or isopropanol, but prolonged exposure may cause crazing or stiffening.
This stands for "other", but the most common is polycarbonate, which is not very soluble in ethanol or isopropanol, but prolonged exposure may cause crazing or stiffening. 


Rubber - most rubbers are probably resistant to alcohols, but prolonged exposure will cause loss of elasticity