Did you escape the $1 in the value? Also, you will need to double evaluate, and fix what you are evaluating:

my %patt1;
my $key = "<date>([A-Z&;][a-z&;]+), ([0-9]{2}) ([0-9]{4})</date>";
my $value = "qq|<month>\$1</month>|";
$patt1{$key} = $value;

$instr = '<date>Jan, 10 1997</date>';
for my $pat (keys %patt1)
{
    if ( $instr =~ s|$pat|$patt1{$pat}|eegs ) {
      print "match1: $instr\n";
    }
}
[download]

If you're doing this repeatedly in your program, though, it would be better to compile the regexes with qr and store them in an array rather than as keys in a hash.