The non-strict (using symbolic references -- not recommended) way:

$val = '123';
while (<DATA>) {
    s/\$(\w+)/$$1/g;
    print;
}

__DATA__
test_string_$val
[download]

The strict-compliant (recommended) way:

use strict;
use warnings;

my %subs = (val => '123');
while (<DATA>) {
    s/\$(\w+)/$subs{$1}/g;
    print;
}

__DATA__
test_string_$val
[download]

Please read Why it's stupid to use a variable as a variable name