Eckersley, 27, has signed until the end of 2018 while Shreck, 38, has signed a one-year contract extension.
Scotland all-rounder Taylor has taken 72 first-class wickets and scored 1,073 red-ball runs for the Foxes since 2011.
Paceman Sheikh's only appearance this season came in the three-day tour match against Sri Lanka in May.
In their draw against Derbyshire in August, Eckersley became only the second Leicestershire batsman to score two first-class hundreds in the same match.
Seam bowler Shreck has played in every County Championship fixture for the county this season, taking 29 wickets.
Leicestershire have given new contracts to Ned Eckersley and Charlie Shreck, though Rob Taylor and Atif Sheikh have not been offered new deals.