open SOURCE, "</path/to/my/file.txt";
my @lines = <SOURCE>;
my $start = <whatever>; # set to where you want to start.
foreach my $ix($start..$#lines) {
   my $line=$lines[$ix];

#Do the get request magic;
}
[download]

another method:

use strict;
use Tie::File;
tie @lines,"Tie::File","< /path/to/my/file.txt" or die "$!";
my $start =<whatever>; 
foreach my $ix($start..$#lines) {
   my $line=$lines[$ix];

#Do the get request magic;
}
[download]

The second method has the advantage of not loading all the file into memory.