Various tests, particularly the client program tests under src/bin, use the Perl TAP tools and are run using the Perl testing program prove. You can pass command-line options to prove by setting the make variable PROVE_FLAGS, for example:
make -C src/bin check PROVE_FLAGS='--timer'
See the manual page of prove for more information.
The make variable PROVE_TESTS can be used to define a whitespace-separated list of paths relative to the Makefile invoking prove to run the specified subset of tests instead of the default t/*.pl. For example:
make check PROVE_TESTS='t/001_test1.pl t/003_test3.pl'
The TAP tests require the Perl module IPC::Run. This module is available from CPAN or an operating system package. They also require PostgreSQL to be configured with the option --enable-tap-tests.