Leaving any required field stops me from proceeding to the next page.
Source: https://tdac.immigration.go.th/arrival-card/#/tac/arrival-card/add
In the example above, I failed to fill in the fields "*City/State of Residence" and "* Phone No.", then clicked on the button "Continue" and got the pop-up message "No data or Insufficient data"
Regarding your question "what fields do you have to leave blank", the answer is "none, but I am free to leave any and all fields without a red asterisk blank.
As another member has suggested, the TDAC site is already up and running for you to try it out, even if you do not intend to arrive in Thailand within the next three days.