From my own personal experience as a smoker, I have to say no we do not nor are the duration of the illness any greather than a non-moker.
One thing I will add to the conversation is that since I retired 5 years ago, the frequency of colds has been reduced by probably 75%. The reason of course is that in an office environment somebody always has something that eventually everyone catches. And since I'm retired I no longer have that close contact with people on a daily basis.
Many people don't give much thought to it but washing your hands when you return home from shopping cuts down on the possibility of picking up cold germs from shopping cart handles and other stuff you might have touched in the marketplace.