Your problem is that you're opening the file for reading, printing out the modified data to STDOUT and then opening it for writing which truncates it. What you probably want is something like this

my $path = "E:/Documents and Settings/Richard Lamb/My Documents/HTML";

open INFILE, "$path/dummy.html"
  or die "$!: Can't open this file";
open OUTFILE, ">$path/new_dummy.html"
  or die "$!: Can't write to the HTML file";

while (<INFILE>)
{
    s/(<h[1-6]\s+style=.*)">/$1; word-spacing: 50px">/ig;
    s/(<li\sstyle=.*)">/$1; word-spacing: 20px">/ig;
    s/(<p style=.*)">/$1; word-spacing: 20px">/ig;
    print OUTFILE $_;
}

close INFILE;
close OUTFILE or die "$0: $!";

rename "$path/new_dummy.html", "$path/dummy.html"
  or die "$0: $!";
[download]

This way you're writing to the output file as you're modifying the data from the input file (not the input file itself) and once that's finished you replace the input file with the output file.

See. open, perlopen and rename for more info on the functions used.
HTH

_________
broquaint