No, there aren't any consequences to dating 5 of either the bachelors/bachelorettes, neither of the cutscenes will trigger, nor are there any cutscenes that suggest something's afoot. Which is too bad, it'd be funny if there was a 5-dating cutscene which didn't really involve you and had no consequences but where someone who's observant could make some comments here n' there, like Gus or Lewis.
There are random consequences to being married while giving gifts to any of the other 5 bachelor/ettes who are the same sex as your spouse. There are no consequences to giving gifts to members of the 6 who are the opposite sex of your spouse. The consequences are temporary and meant to make you feel a bit bad but I won't elaborate. You can still give those 5 people gifts on their birthdays with no consequences so if you're still trying to get everyone to max hearts after marriage, save that loved gift (especially if it can be a starred item) for their birthday.
I just avoid this by treating Stardew Valley as a dating sim, get 5 each to 10

and then get a Rabbit's Foot to avoid the 6-dating cutscene and then date the last of each (looking at
you 
Elliott and

Leah who avoid me!) to 10

before proposing to the unlucky future Mx Farmer Bobbi Cooper.