There is a lot of repetition, most of the if blocks do the same thing and each block repeats its list of headers. And that list of headers it's pretty hard to read. I'd suggest that you look at factoring out much of that duplication. Something like

my %inputfiles = (
    gene2accession => join( "\t",
        qw( Taxon
            GeneID
            Status
            RNA_Nucleotide_Accession
            RNA_Nucleotide_gi
            Protein_Accession
            Protein_gi
            Genomic_Nucleotide_Accession
            Genomic_Nucleotide_gi
            Genomic_Accession_Start_Pos
            Genomic_Accession_End_Pos
            Orientation
            Assembly
        )
    ),
    gene2go => join( "\t",
        qw(
            Taxon
            GeneID
            GO_ID
            Evidence
            Qualifier
            GO_term
            PubMedID
            Category
        )
    ),
    ...
);

foreach my $file ( keys %inputfiles ) {
    ...
    while ( my $line = <INPUT> ) {
        if ( $linecount == 0 ) {
            print OUTFILE "$file\n";
            print SUMMARY "Field lengths for file $file\n";
            print SUMMARY "$inputfiles{$file}\n";
            if ( $file eq 'hiv_interactions' ) {
                # do special processing
            }
        }
    }
    ...
}
[download]

seems to be easier to read. It is obvious that the header is the same in both the summary and out files and that hiv_interactions gets special treatment. The headers are also easier to read. At least something to consider for your next script;)