Slow slide into oblivion
I've heard these types of statements before - they set up such a simple, bipolar worldview. But in many regards I can support this statement in this context. The "die" part is probably overly dramatic, no firm is going to drop dead tomorrow because it does not innovate. However, if your idea of an enjoyable place to work is a place that is a constant follower, afraid to take risks and satisfied with the status quo, then get ready to wither away. I think the more appropriate slogan in this case is "It's better to burn out than fade away". Because that's what firms that fail to innovate will do - fade away.
There are so many competitive pressures on most firms that just keeping up the the latest fads, technologies and new products is daunting enough. Trying to constantly create new products and services is almost overwhelming. However, the market doesn't sit still, and firms that don't innovate will gradually fall further and further behind, becoming a pale shadow of their former selves. Innovation is a mandatory requirement to consistent differentiation and to stand out from the crowd.
Innovate or die? Possibly. But the death will be a long, boring slide into mediocrity and eventually into oblivion. Better to burn out than fade away.