What does a startup absolutely have to do? It isn't creating the most elegantly designed piece of software in the world. It's creating value for the customer.

Along the way to delivering that value, sometimes a less than perfect solution is chosen (often because of resources--time, money, or lack of human resources) OR the perfect solution is chosen, but the product changes and that solution no longer is perfect.

Technical debt should be managed, but it can't be eliminated... unless you want to give up producing anything new, and just work on polishing your previous work forever.

When will it kill your startup? If it turns out that your product is NOT what your customer actually wants, but the shortcuts (tech debt) you took prevent you from making the necessary adjustments to meet your customers needs.