It seems to work for me. This code enters the while loop and prints out the value of the input file:

> cat last_assignee.txt
john
> ./684200.pl
In while loop
john
> cat 684200.pl
#!/usr/bin/env perl

#use warnings;
#use strict;

open( last_assignee, '<', "last_assignee.txt" ) or die "Unable to open
+ last_assignee.txt for READ\n";

# Get the last assigned value and store it to a variable
while (<last_assignee>) {
    print "In while loop\n";
    print $_;
    chop;
    if ( $_ ne "" ) {
        $assignee_last = $_;
    }
}
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Are you sure your input file has anything in it?

I intentionally commented out use warnings; use strict; just to try to duplicate your results. Make sure you use these in your code.