When did mouth pipetting stop becoming a way to handle liquids in a lab?
Mouth pipetting, while almost unheard of in modern laboratories in developed countries, is still very much a current protocol in many parts of the world. 

For example, this paper analyses the proportion of clinical labs in Pakistan and found evidence of poor biosafety practices (emphasis mine):


  Results: A total of 1,647 (92.4%) males and 135 (7.6%) females
  participated in the study, with over half (59.7%) having more than
  five years of work experience. Results showed that 28.4% of the
  laboratory technicians from Punjab, 35.7% from Sindh, 32% from
  Balochistan and 38.4% from Khyber Pakhtoon Khawa (KPK) did not use any
  personal protective equipment. Almost 46% of the respondents (34.2%
  from Punjab, 61.9% from Sindh, 25.2% from Balochistan and 85% from
  KPK) said they reused syringes either occasionally or regularly.
  Furthermore, 30.7% of the respondents said they discard used syringes
  directly into municipal dustbins. The majority (66.7%) claimed there
  are no separate bins for sharps, so they throw these in municipal
  dustbins. Mouth pipetting was reported by 28.3% technicians.


Additionally, a Google Scholar search for papers published in the past 5 years involving mouth pipetting turned up 670 hits out of 284000 total hits, a proportion of 0.23%. This is further evidence that this dangerous technique has not been fully eradicated from labs.