Thank you, sir. I've incorporated your advice into my code in conjunction with advice about how I open my file handle and it seems to be working. Here's how my code looks now:

my %id2seq = ();
my $id = '';

open File,"human_hg19_circRNAs_putative_spliced_sequence.fa",or die $!
+;

    while(<File>){
        chomp;
        if($_ =~ /^>(.+)/){
            $id = $1;
        }else{
            $id2seq{$id} .= $_;
        }
    }


    foreach $id (keys %id2seq){
        open  my $out_fh, '>>', $id or die $!; ##Amendment here
        print $out_fh ($id."\n",$id2seq{$id}, "\n");
        close $out_fh;   ## moved into the foreach loop
    }


close File;
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