I might slightly rewrite as follows:

open my $in, '<', 'abc.xml'
    or die "Can't read abc.xml: $!";
open my $out, '>', 'iabc.xml'
    or die "Can't write iabc.xml: $!";
while ( my $line = <$in> ) {
    $line =~ s/\\limits//g;
    print $out $line;
}
close $in  or die "close input: $!";
close $out or die "close output: $!";
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I note that the string you say you're looking for ("\limits") is not in the example string you give ("$ a+b=\sum\limit_a^{n-j}$"). I also note that the example string appears to be TeX, so a non-Perl solution might be to redefine the macro.