$query =~ s/$value/$replace/i;

The problem is that $value contains something like $Id, and the $ when interpreted as a regex means "match the end of the string", not "match a dollar sign". You can fix that by writing

s/\Q$value\E/$replace/i;
[download]

instead.

Fwiw this is how I'd solve the same task:

my @array_1 = ("\$Id_run_query=1", "\$name_run_query=xyz", "db_query_r
+un_query=select * from meter where metetid=\$Id and metername=\$name"
+);
my $query;
my %replace;

foreach(@array_1){
    if (m/^db_query_run_query=(.*)/i){
        $query = $1
    }
    elsif (/(\$\S+)_run_query=(.*)/) {
        $replace{$1} = $2;
    }
}
my $replace_re = join '|', map quotemeta, keys %replace;
$query =~ s/($replace_re)/$replace{$1}/ig;

print "===>$query\n";
[download]

The call to quotemeta does the same as \Q...\E in the regex.