"Blowing smoke up someone's arse" is a figure of speech that has its origin in the tobacco smoke enemas that were popular during the 1700s and was used to treat a variety of health problems ranging from headaches to respiratory illness and as a means of resuscitation victims of drowning as it was thought to "dry out" the victim's insides.
According to an article published in historycollection.com, Robert Mead was the first known Westerner to suggest tobacco smoke enemas as an effective treatment for resuscitation in 1745. It was thought to provide benefits as a respiratory stimulant and to be used for warming up the human body. Two doctors who worked at The Institution for Affording Immediate Relief to Persons Apparently Dead From Drowning, published a rhyme to help the public successfully perform the procedure:
Tobacco glyster, breathe and bleed.
Keep warm and rub till you succeed.
And spare no pains for what you do;
May one day be repaid to you.
The article goes on to explain that the tools for this procedure included a tube, a fumigator, and bellows. One end of the tube was inserted into the victim's rectum and the bellows pumped to send smoke into the person's body. In the event that all the tools were not readily available, a tobacco pipe has been known to have been used. This sometimes led to fatal consequences for the person doing the blowing as a result of respiration of fecal matter especially if the victim was suffering from an infectious disease.
An article that was published by allthatisintersting.com talks about how the 1811 discovery of the toxicity of tobacco to the cardiac system reduced the popularity of smoke enemas.