while (<>) {
    # $_ holds the line read - process it here
}

##</code><code>##

... (keys ...

##</code><code>##

... (sort {length $sequences{$a} <=> length $sequences{$b} } keys ...

##</code><code>##

$ cat pm_1144803_fasta.txt
>SequenceID|1234_Gene1
CTTTTAAGCTGATTAGGCTTTTATACCATTAGATTTAGTAACTATTGTCTTTTAA
>SequenceID|9876_Gene2
GTGCTGTCTTAAGTTGAACAGAGTGTGGGAGGAAATATAAGCAAAGTTATTCCGTAGAATT
>SequenceID|4567_Gene3
CATCCTCCTTTACACCCCACAAACATTTGGCAACCCCTGATAGGTTTCTTTCTTGTGGA

##</code><code>##

$ cat pm_1144803_fasta_seq_len_sort.pl
#!/usr/bin/env perl

use strict;
use warnings;

my (%sequences, $seq_key);

while (<>) {
    chomp;

    if (/^>/) {
        $seq_key = substr $_, 1;
    }
    else {
        $sequences{$seq_key} = $_;
    }
}

foreach my $key (
    sort {length $sequences{$a} <=> length $sequences{$b} } keys %sequences)
{
    my $len = length ($sequences{$key});
    print "$key:$len\n";
}

##</code><code>##

$ pm_1144803_fasta_seq_len_sort.pl pm_1144803_fasta.txt
SequenceID|1234_Gene1:55
SequenceID|4567_Gene3:59
SequenceID|9876_Gene2:61