A couple suggestions:

1. If you are just incrementing by 1, use Foreach Loops instead of C-style loops -- fewer moving parts:

   for my $i (0 .. @F-1) {
   [download]

2. Avoid punctuation variables if you can, unless this is your toy and you are super comfy with them. Rather than testing $., just test if @F1 is initialized:

       @F1 = @F if !@F1;
   [download]

   Note that you shouldn't use logical compound assignment operators because they have scalar context.
3. Contrasting the above, you should be using $_ in this case because $line is so highly localized.

   while (<DATA>) {
       chomp;
       my @F = split '&';
   [download]

4. +@F1 is a no-op. Numification requires two arguments, so you'd need to write 0+@F1, but you don't even need to do that because logical operators like != also apply scalar context to their arguments.
5. Your algorithm gets simpler and allows using a hash if you track which terms to delete rather than which ones to keep. I'm assuming that you don't have repeated keys.

So I might write that as:

#!/usr/bin/env perl -w
use v5.014;
my %seen;
my $count;
my @recs;
while (<DATA>) {
    chomp;
    my @F = split '&';
    $count //= @F;
    die "NF mismatch" if @F != $count;
    $seen{$_}++ for @F;
    push @recs, \@F;
}

for my $rec (@recs) {
    say join "\t",
        grep $seen{$_} != @recs, # Doesn't show up in every record
        @$rec
        ;
}



__DATA__
a=1&b=1&c=1&d=2&e=&f=3
a=1&b=2&c=3&d=2&e=&f=4
a=1&b=2&c=5&d=1&e=&f=5
[download]

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