I've set LOOKUP_INCLUDE=-I/client/include in Local/Makefile, but the compilation of exim_dumpdb is ignoring this option and failing. Why?

LOOKUP_INCLUDE is the special include file for lookup modules in Exim (e.g. mysql, LDAP). Confusingly, it doesn't apply to basic DBM code which is used also for other things. Try setting INCLUDE and DBMLIB instead. For example:

USE_DB=yes
INCLUDE=-I/client/include
DBMLIB=/client/lib/libdb.a