There are a few things I would do differently

• I'd put the most of the variables into the loop, restricting the scope to just the places they are used will spare you some grief later.
• You need to test that your regular expression matched, if it doesn't you end up processing the previous record again.
• If you have a student with the surname O'Hara you will fail to process her records.
• $status has the value of the last mark checked, I think you mean to check if any failed
• The use of . to join the strings is ugly, use variable interpolation instead.

Anyway this is my version, it still fails to report marks for Ms. O'Hara but at least you will get a warning, and I have to leave something for you;-) The other change is to not explicitly code the number of tests, this will work if the input file has results of four tests.

#!/net/perl/5.10.0/bin/perl

use strict;
use warnings;
use List::Util qw(sum);

my $input_file = 'marks.txt';

open my $DATA, '<', $input_file or die "Can't open $input_file: $!\n";

while ( my $line = <$DATA> ) {
    next if $line =~ /^NAME/;
    chomp $line;
    if ( $line =~ /^\w+,\w+(?:,\d+)+$/ ) {
        my ( $first_name, $last_name, @marks ) = split ',', $line;
        my $totalmarks = sum @marks;
        my $average    = sprintf "%.2f", $totalmarks / @marks;
        my $fail       = 0;
        foreach my $mark (@marks) {
            $fail ||= checkfail($mark);
        }
        if ( !$fail ) {
            print "$first_name $last_name\t@marks \t$totalmarks\t$aver
+age\n";
        }
        else {
            print "$first_name $last_name\t@marks \t\n";
        }
    }
    else {
        warn "Invalid input: $line\n";
    }
}

close $DATA or die "Can't close $input_file: $!\n";

#To check whether the student failed
sub checkfail {
    my $marks = shift;
    return $marks < 35;
}
john michael    100 50 60     210    70.00
sam shane    50 60 70     180    60.00
tom christo    30 40 50
[download]

You may wish to give consideration to using Text::CSV_XS for parsing comma delimited text files.

Update: two small code modifications kindly suggested by moritz.