Since your data structure needs to accommodate dates with and without a time, you have two choices:
Use a Ruby DateTime object with a flag value for the time to indicate that the date does not have a time. The usual flag value for this is 0 which then means the midnight time can't be shown. (Midnight is 0 seconds after the day has started.)
For example, parsing "Jan 1, 2010" into a DateTime will give you Jan 1, 2010 00:00.
Otherwise you'll need to invent your own data structure. Easiest would probably a Class with a DateTime and a "show_time" boolean flag. -- by using a DateTime to hold the time, you'll be able to use the DateTime output methods, and do arithmetic with them if needed.
Creating a new data structure is not such a big deal in Ruby, but if you can live without tasks due exactly at midnight, I'd recommend method 1. Note that you'd probably want to print them without a time since that's what the task definer requested. Or you could include "(any time)" in the output.
PS
Watch out for timezones! Many ways to handle, but you should be sure to choose one deliberately.