Will a Fields form with many submissions make an overload of nodes? I need to create some user submission forms, and I'm debating the merits of using a custom content type Fields API forms, or a custom module form like the ones in Examples module.

As I understand it, Fields API forms will create a new node for each submission. These forms will have hundreds of submissions every day all year long, and this will create a lot of nodes. I am wondering if this would present a problem in terms of database performance, and site management.
Generally, the bigger the database the potentially slower it can get. If you get 200 nodes submitted every day that's 73,000 nodes a year. A lot of news sites or forums or w/e that run on Drupal easily surpass numbers like that. 

You can use the Form API to capture content in a custom table of your creation, there's absolutely nothing wrong with that. If you need everything that comes along with nodes, fields, contrib modules that plug into nodes, etc., than you may want to just use nodes. Then there's the ability in D7 to just create a custom fieldable entity (not a node but "like" a node in that it's a piece of content that Views and other things can hook into) and save overhead by keeping things lean.