What happens to a company whose share price goes down to very little over dividend loss? This is a bit of an edge case scenario.

Let's say you have a company that is very profitable (and pays very high dividends off the profit) but has little/no growth potential.

The company decides to slash the dividends significantly (to the point that the total yield is super low, lower than most other equity) in order to pay significantly higher wages/benefits to their headcount.

Presumably, the share price of the company would plummet drastically, as nobody would want to own the shares (zero dividend + zero growth means you're better off investing in a savings account, at least outside of negative interest rate countries).

What happens to that company's finances longer term? 

Their valuation just shrunk to almost nothing, but they aren't bankrupt as they still have the same cash flow, same same business, same outstanding debt and same profits as before - they merely redirected the profits from dividends to headcount costs.

(let's ignore real-world implications such that the company may start experiencing growth out of better publicity, employees using extra earnings to purchase their products ala Henry Ford-UAW parable, etc...).
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.