Another way, using a positive look-ahead assertion and the  \K special escape of 5.10+ (see  (?<=pattern) \K in Look Around Assertions) which is effectively a variable-width positive look-behind (an actual look-behind in Perl regex may only be fixed-width):

>perl -wMstrict -le
"my $s =
   'there is no FOO stuff i do not want BAR unwanted BAR stuff here';
 ;;
 my $t = $s;
 $t =~ s{ FOO \K .*? (?= BAR) }{}xms;
 print qq{'$t'};
 ;;
 $t = $s;
 $t =~ s{ FOO \K .* (?= BAR) }{}xms;
 print qq{'$t'};
"
'there is no FOOBAR unwanted BAR stuff here'
'there is no FOOBAR stuff here'
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The given examples illustrate the effect of the  ? quantifier modifier to achieve 'lazy' matching. If the substitution is to be done repeatedly, the  /g regex modifier should be used.

As suggested above, a positive look-behind could have been used in the examples because the pattern in question is fixed-width, but look-behind assertions, positive or negative, cannot be used for general patterns, which seemed to be the thrust of the OP.