I cleaned up your code and it now looks as follows:

use strict;
use warnings;

my $filename = 'fasta.txt';

open my $FILE, '<', $filename or die "Could not open $filename: $!";

while(my $line = <$FILE>) {
    if($line=~/^>.*unknown/ ) {
        $line .= <DATA>;
        print $line;
    }
}
close $FILE;
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How do you know your script stopped reading after finding the first header? Are you sure your file contains more than one header?

I ran the above on the following data:

test
not this
> unknown
print this
no no
no again
> known (not this header)
got you!
no way Jose
> HEADER unknown
Yessir!
not this line
throw this away
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Output:

> unknown
print this
> HEADER unknown
Yessir!
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