A couple of comments on your code:

• If the word 'ORIGIN' appears anywhere else in the test, your code would break.
• If you use $/='ORIGIN' your file would automatically be split at all occurences of this word and you could just use the last bit.
• Instead of removing all kinds of unwanted characters you could tell Perl to remove everything but a, c, g, and t.

Most of it is clearly a matter of taste but it feels more direct to me this way:

use strict;
use autodie;

open my $fh, '<', 'fluseq.txt';
my @tmp = do {local $/='ORIGIN'; <$fh>};
my $dna = pop @tmp;
$dna =~ s/[^acgt]//gi; # delete all but a, c, g, and t
print $dna;
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