Update:
Roger pointed out some gross errors. Sorry! Fixed now!

Well, other than a few errors, you're on your way. I'll do a line by line on this baby and see what I think.

my $file = "index.html";

my @codes = ("<P>", "<BR>", "UL"); # and many more
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Do you mean to be inconsistent here? Or should "UL" be "<UL>"?

open(FILE, "> $file") or die "Oops: $!";
while(<FILE>);
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Your open is clobbering index.html. Not what you want. Also, you never write out to the file to actually change it in memory.
Try  while (<FILE>) { :)

if ($_ eq @words) {
$_ =~ s/@words/lc @word/g;
}
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Here's where you have your real problems. The if statement compares the file line to the number of items in @codes. Probably not what you want. Also, eq won't work unless you have only one word per line. Also probably not what you want.

close(FILE);
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Way to go.

I would rewrite it like this:

#!/usr/bin/perl -i
use warnings;
use strict;


my @codes = ("<P>", "<BR>", "<UL>"); # and many more


while(<>);
for my $word (@words) {
$_ =~ s/$word/lc $word/g;
}
close(FILE);
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Call it with the filename as argument.
Of course, this is still horribly simple minded because it breaks on constructs like <P class="hi">.

Good luck fixing that with a simple regex.