You can use the tell() function to get the current file offset, and seek() function to get back to that offset, once you've reopened the file.

use strict;

## obtain $LastPosition

open(LOG, '/var/log/messages');
seek(LOG, $LastPosition, 0);


while (my $line = <LOG>) {
    chomp $line;
    ## Process $line;
}

$LastPosition = tell(LOG);
close(LOG);

## store $LastPosition
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