You need to add the modifiers at the start of the substitution like this:

use strict;

my $pattern     = 'test';
my $replacement = 'New';
my $flags        = 'i';

my $value        = 'My Test Text';

$value             =~ s/(?$flags)$pattern/$replacement/;

print "$value\n";
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This will print:

My New Test
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