Sign up page for one product

Assign negative price group for that product, it will be removed from default signup page and will become available by special url: /amember/signup.php?price_group=-1 (for example)
Then edit signup template change this line:
{if $config.use_address_info}

to
{if $config.use_address_info && $smarty.request.price_group <> -1}

Above will remove address info block.

Delete them from template, or if you want to remove these fields from price_group=-1 signup page apply these tags around these fields:
{if $smarty.request.price_group <> -1}
{/if}
Then remove validation of these fields from signup.php:
if (!strlen($vars['name_f'])){
$error[] = _SIGNUP_PLEASE_ENTER_FNAME;
}
if (!strlen($vars['name_l'])){
$error[] = _SIGNUP_PLEASE_ENTER_LNAME;
}