Google Sheets – Automatic Date Insertion with Zapier

This answer is coming late to the party, but you can achieve what you want by installing this script:

function onEdit(event) {
  var eventRange = event.range;
  if (eventRange.getColumn() == 4) { // 4 == column D
    var columnARange = SpreadsheetApp.getActiveSheet().getRange(eventRange.getRow(), 1, eventRange.getNumRows(), 1);
    var values = columnARange.getValues();
    for (var i = 0; i < values.length; i++) {
       values[i][0] = new Date();
    }
    columnARange.setValues(values);
  }
}

EDIT: It seems you just want to trigger the script when column D is modified, so I have modified the code accordingly.

To install it, go to Tools -> Script editor, paste the code above, and save. From then on, every time you edit a cell in column D, the current timestamp will be inserted in column A.

Using the Trello API, I created a bot that can do exactly what you want (and a lot more). It's been running for a while with very satisfied users (see this reddit thread).

To use it, you invite user "butlerbot" to your board. It will create a list called "Butler" automatically. You enter commands as cards in that list.

The command you want is something like:

on a card's due date, move it to list "Inbox"

This command will move the card at midnight the day it's due. If you prefer a different time, you can specify it:

on a card's due date at 8pm, move it to list "Inbox"

(You may want to set your timezone if you're not in US eastern time, there's a simple command for that too.)

Another alternative:

the day after a card is due, move it to list "Inbox"

This will move it to the list the midnight after its due date.

If you need help using it, there's a public Trello board. The developer (that would be me) is very responsive.