In the worst case, index will crawl the entirety of $str_a looking for the character you extracted from $str_b so it can't be the fastest solution.

You want to make sure you only make a single one-character comparison. And because one of the strings is invariant during the loop, you want to move its first-character extraction out of the loop. That gives us the canonical solution of

my $str_a = 'Foo';

my $chr_a = substr $str_a, 0, 1;

for my $str_b ( @list ) {
    next if $chr_a eq substr($str_b, 0, 1);
    expensive_function($str_a , $str_b);
}
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Update: beaten to the punch by kvale.