davorg has correctly identified where the bug lies. 'if ( $first = 1 )' will assign 1 to $first and then submit the same as the value of the 'if' condition (1 being a true value). But there are some things which should become habitual, even when writing the first version:

#!/usr/bin/perl
use strict; # avoid typos and unclear scoping...
use warnings; # ...amongst other things
# open fhand,file.txt; # this will now cause the
                       # compilation error that it should
                       # ...replacement further down...
# declare and init your vars, including file handles
my $file = 'file.txt'
my $path = $ENV{ SOME_PATH } . '/' . $file; # or must cd?
open my $fh, "<$path"
    or warn "$!: $file\n" and exit $?;
    # assume I/O can and will go wrong sometimes
my $first = 0;
while( <$fh> ){
    # indent your nests for clarity
    # avoid falling thru from one case into the next
    if ( /First Name/ ) {
        $first = 1;
        # any other processing of First name line
    }
    elseif ( /Last Name/ ) {
        $first or die "Order is not correct\n";
        # any other processing of Last name line
    }
    elsif ( 0 ) { # ...etc....
    else {
        # default or 'all other' case if applicable
    }
}
close $fh; # specifically close filehandles
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