^ means normally means "beginning of string", not "beginning of line". The m modifier changes that. (See perlre.) That gives you two options:

$TheFile =~ s/^'//mg;
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and

while (my $TheLine = <>) {
   $TheLine =~ s/^'//;
   ...
}
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Update: You probably want to use '.* instead of just ' unless you only want to delete the '.