Answer
Most do. IIM Lucknow requires 85 in each of VARC, DILR and QA; IIM Ahmedabad requires 85 in each; IIM Visakhapatnam requires 70. Three IIMs namely Rohtak, Sambalpur and Sirmaur publish no sectional cut-off. Nearly all IIMs additionally require a positive raw score in every section, so a blank or negative section disqualifies you regardless of overall percentile.
Question
Do IIMs have sectional cut-offs?