Does API limit affect published forms? In what way? Thank you!

The API Limit will not directly affect your published form. The API limit will only affect the number of times that you can call/use the Jotform API each day.

For example, you are using the Jotform API to connect your form with a third party application. Every time a user makes a submission, an API call will be made so that the submission will be forwarded to the third party application.

Each call made by the API will count towards your API limit. Free users can make 1,000 calls per day. Once you reach that limit, you will no longer be able to make an API call for the rest of that day. Please note that the API limits counter will reset every midnight EST.

You can find out more information about the Jotform API here: http://api.jotform.com/docs/#overview

If you need further assistance, please let us know.
Regards.