What am I doing wrong?

• Your glob may not match any files
• Your script may not be opening any matched files
• Your grep regex may not match what you want it to
• Your unlink may be failling

That's just 4 possibilities and you won't know which is the real problem until you add some diagnostics to your script. eg (untested):

use strict;
use warnings;

while ( my $file = glob '*.txt' ) { 

    print "Debug: found file $file\n";
    open my $fh, '<', $file or die "Cannot open file $file: $!";
    my $headers = grep /^>/, <$fh>;

    if ($headers <= 3) {
        unlink $file or die "Cannot unlink file $file: $!";
    } else {
        print "Debug: file $file only has $headers headers - skipping\
+n";
    }
}
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