Yes, every task can have a due date and time. If you're adding a task (or editing it) simply hit the TAB key to get to the due date field in the bottom-right corner of the task input field and you can type: "tomorrow at 2pm" or "ev day at 13:10" within this field to set a date/time for a task.
For a full list of all supported date formats, please refer to this page: https://todoist.com/Help/timeInsert
I don't think this is working properly for me, I can set days but not times -- I get a large blue box showing up on my screen when I try to drill into the day to set any time frames.
Please confirm - you use the web-app at https://todoist.com, you go to the due date field as you add or edit a task and when you type: "tomorrow at 2pm" in this field... what happens exactly?
Could you please send a screenshot to firstname.lastname@example.org and email@example.com showing what happens when you try to add such due date?