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

... (keys ...

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

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

$ cat pm_1144803_fasta_seq_len_sort.pl
#!/usr/bin/env perl
...
    my $len = length ($sequences{$key});
    print "$key:$len\n";
}

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