How can I get Siri integration with Remember the Milk to add items for a particular date?
The word 'today' doesn't register with Siri and reminders.  Reading through the documentation on Apple's support site I found this:


  
  If you enter “breakfast” or “morning,” Reminders sets a 9 a.m. notification.
  If you enter “lunch “ or “noon,” Reminders sets a 12 p.m. notification.
  If you enter “dinner” or “night,” Reminders sets an 8 p.m. notification.
  If you enter a day, but don’t enter a time, Reminders sets a 12 p.m. notification.
  If you enter a time, but don’t enter a day, Reminders sets a notification for today.
  


It doesn't mention Siri, yet after some testing the rules seem close to the same for Siri input, although the times don't match. 

Solutions:


Give Siri a time without a date. 


  "Remind me to go to the dry cleaners at 5pm"

Give Siri a duration


  "Remind me to go to the dry cleaners in 2 hours"

Use "evening" and Siri will set the reminder for 6pm


  "Remind me to go to the dry cleaners this evening"

Use "tonight". Siri will set the reminder for 7pm


  "Remind me to go to the dry cleaners tonight"

Use "noon" and Siri will set the reminder for 12pm


  "Remind me to go to the dry cleaners at noon"

Use "morning" and Siri will set the reminder for 7am


  "Remind me to go to the dry cleaners in the morning"



Last note, if you are past the time that the word represents Siri gets confused.    Asking "this evening" after 6pm will cause Siri to ask you when you'd like to be reminded.