The benefit of testing is confidence that things are working. There are side benefits, like acting as documentation and examples, but they should not be the focus.
Writing tests (or deleting them, refactoring them, etc.) should always involve a cost/benefit calculation, even if it's a rough mental estimate that's not written down. In particular, that requires answering "How much effort will this test take me to write/debug?", "How much extra confidence will having this test give me?", "What level of confidence do I feel comfortable with?" and "Could I achieve higher confidence spending this effort on something else?".
This doesn't need much effort. For example, we might think "This array indexing took me a few attempts to get right; it could result in misleading reports, so I'd better throw a few edge cases at it to make sure I understand it correctly.". On the other hand we might think "This class doesn't have any unit tests, but there's already a bunch of validation on the results; it only affects the page layout, and it'll be obvious if something's wrong, so I'll leave it for now and put some extra checks on the gnarly payment system."
This approach makes sense regardless of the situation. In your startup example, it might make a lot of sense to write some tests that, for example, spider our site for broken links; or look for certain strings on certain pages. Things which are quick to write (they could even be shell scripts), but give us prior warning that our demo will go wrong. It would make less sense to write a pile of unit tests for the internals of some boilerplate piece of the system.
Code coverage, dogma, etc. should not be the deciding factors for what to work on.