Yes, perseverance is important (particularly for undergrads, who have probably never worked on a project with a time horizon longer than a semester), but you can't fix burnout just by lowering your head and trudging forward like a packhorse.
I don't know if there's a single, quick fix for burnout, but taking a vacation is definitely a good idea -- even if it's only a few days. And if you can't take your mind off of what you're doing for a few days, something else is wrong.
Usually when people say "I can't take a break" they're engaging in catastrophic thinking. If for some reason they really can't take a break, it's time to take a good, hard look at how you're doing things, and see what it is that's causing that situation. In my experience, the root to your burnout will often lie in an honest exploration of that question, alone.
It's incredibly helpful to get an outside perspective on your situation when you feel this way -- this is what counseling is for. If you can't afford that, talk honestly with a friend you can trust -- a friend who has no personal interest in the outcome. That's critical. If you ask your co-founder what you should do, you have to realize that s/he'll be biased.