A Minimum Viable Product is used to validate whether your idea has legs. It should be released early enough that you should be embarrassed about it. So how well tested should it be? And by this I mean unit/integration tests, not a manual click through test.
A safe estimate for including tests is to double your development time. Theoretically, you could pump out two untested MVPs versus one fully tested MVP. However, if your MVP has legs, you have an untested product which slows you down in the long run.
Three scenarios off the top of my head (not exhaustive):
1. Untested. If it works out, add tests after. You can pump out MVPs quickly, but will be slow afterwards.
2. Fully tested. If it works out, you can move quickly afterwards, but will take longer to reach validation.
3. Half-tested. Best of both worlds? However, modifying existing tests feel just as hard (or harder) than starting from scratch. Could be worst of both worlds?
Should you go for speed or robustness? What has worked for you?
Are you launching hot-or-not_2.0? Well, there's not a lot than can go really wrong if it totally burns.
Are you launching mint-for-the-enterprise? Well, you might not get a 2nd chance if you accidentally transfer all of a users funds to Nigeria as part of your automated email handling code.