EDIT: This is a method to remove a single time from an events list without removing all times from the list. This is not how you would hide the time from all events.
I believe I have this solved (somewhat), however, it requires jQuery. It works by using a "unique" time to target and then add a custom class to it. Then, we hide that class and thus, hide the time. The custom time I've used is 1:23am -- so any event you want to hide the time on, you set the start time to that value. Unfortunately, if you have an event that starts at that exact time, this won't work.
This is for the FIVE template, I'm not positive if it works for other templates.
To install jQuery to your site, go to 'Manage Site' (gear icon) > 'Settings' > 'Code Injection' and paste this into the 'Header'
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
Next, go to your Events page, and go to 'Events Settings' > 'Advanced' and paste this into the 'Page Header Code Injection'
<script>
$( document ).ready(function() {
$('.eventlist-meta-time:contains("1:23am")').addClass('hideTime');
});
</script>
Lastly, go to your Custom CSS section, and paste this:
.hideTime { display: none !important; }
Now, as I mentioned above, in order for this to work, the event start time must match the value in the 'contains(" ") -- this can be changed to whatever time you want, but make sure you set the times of the events to match.
Another thing to consider is that the time might show up in other places -- I have turned off the 'Event Date Label Time' in the 'Style Editor' (under 'Event Styles').
Another place where the time will show up is in the 'Calendar block' -- I decided to hide this for all events (aesthetically, I prefer them hidden) -- to do this, you would add this to your custom CSS:
.sqs-block.calendar-block .yui3-calendar-day .event-menu .event-time-group .event-time { display: none; }
This isn't a perfect solution, obviously, and I hope SQSP adds a 'hide time' and 'all day' event options -- as well as 'repeating events'.