What happens to a company whose share price goes down to very little over dividend loss?
What happens is completely dependent on the owners. They're the ones whose income has been taken anyway. They may be the only ones with the legal power to do anything (depending on the jurisdiction: there may be some countries where the State can intervene in such matters).

In some jurisdictions, the directors have a legal obligation to maximise returns to the shareholders over the long term; in other cases, they may have a contractual obligation to do so. And in most cases, that will be the expectation of the shareholders, anyway. They can intervene at an Annual General Meeting or an Extraordinary General Meeting: they can pass motions to change future dividend policy; or replace the directors - they could, of course just become the board of directors themselves. And if there's a legal or contractual obligation, then there may be recourse available through legal action too, to seek compensation for the directors' past transgressions.

If they decide to do nothing, then nothing in particular happens to the company's finances.