While the above are good answers, there is an ambiguity in the spec. What result would you expect for:

my $left_text   = [ qw{ when rome romans} ];
my $right_text  = 'When in Rome, Romans do as the Romans do in Rome.';
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Would you expect a result of 3 or of 5 (or something else)? Essentially, do you want a count of all case-insensitive matches or just a count of how many search words matched? If you want the latter, the following would suit your purposes better:

#!/usr/bin/perl
use strict;
use warnings;

my @left_text   = qw{ when rome romans};
my $right_text  = 'When in Rome, Romans do as the Romans do in Rome.';

my $count = '0';
for my $search_text (@left_text) {
    if ($right_text =~ /\Q$search_text\E/i) {
        $count++
    }
}
print "Found = $count\n";
__END__
Found = 3
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Update: And while you're at it, what do you expect for

my $left_text   = [ qw{ when rome romans} ];
my $right_text  = 'When in the Terrordrome, do as the Romans.';
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If you would only expect 2 matches (ignoring that "rome" is in Terrordrome), then you will want word boundary assertions (\b):

#!/usr/bin/perl
use strict;
use warnings;

my @left_text   = qw{ when rome romans};
my $right_text  = 'When in the Terrordrome, do as the Romans.';

my $count = '0';
for my $search_text (@left_text) {
    if ($right_text =~ /\b\Q$search_text\E\b/i) {
        $count++
    }
}
print "Found = $count\n";
__END__
Found = 2
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See perlretut or perlre for more info.