You will need to keep a count of the total per row, but you are not doing so yet. Further your code is trying to add a single value to an array which logically makes no sense. Also, each row contains the gene name as well as the data so you need to avoid trying to add a number to a string.

Using your loop structures then we might try this instead:

for($row = 0; $row < $no_of_seq; $row++){
        my $sum = 0;
        for($col = 1; $col < $no_of_seq; $col++){ 
            $sum += $myArray[$row][$col];
        }
        print "$myArray[$row][0] $sum\n";
}
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But that's not very perlish. Let's improve the loops:

for my $row (@myArray) {
        my $sum = 0;
        for ($col = 1 .. $no_of_seq - 1) { 
            $sum += $row[$col];
        }
        print "$row[0] $sum\n";
}
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This could be further cleaned if you work on a copy of @myArray which you could disrupt as you go. Putting this all together we arrive at the SSCCE:

#!/usr/bin/env perl 
use strict;
use warnings;

my @myArray;

while (<DATA>) {
    my @columns = split /\s+/, $_;
    push @myArray, \@columns;
}
my $title_row = shift @myArray;

for my $row (@myArray) {
    my $sum = 0;
    for my $col (1 .. $#$row) {
        $sum += $row->[$col];
    }
    print "$row->[0] $sum\n";
}

__DATA__
GeneID    Tp1    Tp2    Tp3        
ALA1    10    12    11        
THR8    57    99    12        
HUA4    100    177    199        
ABA5    2    5    10
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