Two things:

• You never saved anything back to your @links.
• There was no need to create that $uri variable unless you need it somewhere else.

use Data::Dumper;
use URI;

use strict;
use warnings;

my $base_url = "http://www.abc.com/";
my @links = qw/a b c/;

for my $index (0 .. $#links) { 
    $links[$index] = URI->new_abs($links[$index], $base_url)->as_strin
+g;
}
print Dumper(\@links);
[download]

You can use map instead of for loop:

use Data::Dumper;
use URI;

use strict;
use warnings;

my $base_url = "http://www.abc.com/";
my @links = qw/a b c/;

@links = map {URI->new_abs($_, $base_url)->as_string()} @links;
print Dumper(\@links);
[download]

Both gives:

$VAR1 = [
          'http://www.abc.com/a',
          'http://www.abc.com/b',
          'http://www.abc.com/c'
        ];
[download]

Your code doesn't seem to do what you want it to do...

This begs for a map, as seen in this untested code:

my @newlinks = map { URI->new_abs( $_, $base_url ) } @links;
[download]

You're assigning to the wrong variable. Change
foreach (@links) { $uri = URI->new_abs( $_, $base_url ) }
to
foreach (@links) { $_ = URI->new_abs( $_, $base_url ) }</p