Consider the following:

use IO::File ();

##  local $/;  # you're not using this

my $fh = IO::File->new();  # declare this just once

foreach $file (sort keys %altnames) { 
   my ($hr_title) = ( $file =~ m/(.*)\.txt/ );
   print "<hr> <h1 align=center> $hr_title </h1>\n";
                # don't need parens for print ...

   $fh->open( "$dir/$file" ) or die $!; # you gotta check!
   while (<$fh>) {
      print;
   }
   $fh->close;
}
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I think that having only one instance of IO::File will help (but I haven't tested this). Also, having something besides $_ as the "foreach" loop variable could make a difference, since you are also using $_ to iterate in the "while" loop, which might cause unexpected consequences (e.g. changing the strings that are used as the keys in %altnames).

update: I rigged up five dummy files with names as indicated in your error messages, and tested your original code with perl 5.8.0 on linux SuSE 7.3 -- didn't get any errors, so I'm at a loss. The comment about altering the hash key strings in the while loop was misguided; still, it seems better to have the outer loop using an iterator variable that's different from the inner loop. Anyway, my version ran as well (after I fixed a missing dash character in the "IO::File->new()" line).