Probably the most well-known example of this is that POSIX states that filesystem blocks are 512 bytes per block, so the GNU fileutils such as df(1) and GNU tar(1) use 512 if the variable POSIXLY_CORRECT is set, and 1024 bytes per block by default. (See Democracy Triumphs in Disk Units)
Many of the GNU tools comply with POSIX by default, except for where the author thinks the POSIX standard is wrong or dumb. :) For the same reason, some programs treat an variable named POSIX_ME_HARDER as an acceptable alias for POSIXLY_CORRECT.
Some programs that behave differently if POSIXLY_CORRECT is set: