Stating the obvious part first: While you are in debt, you cannot hire new staff, purchase new rooms, or purchase new objects, fund research, or begin a marketing campaign.
So really, if you are in debt, there are no more loans available, and you aren't making money, you would be forced to restart to make any progress in most scenarios. The game makers realized this and implemented some anti-stalemate code:
Once your debt hits $150,000 or more, the advisor will give you an ultimatum: If it's allowed to go beyond $300,000, you lose immediately. (-$300,001 balance == game over)
Note that these numbers are hard-coded, and rather unrealistically completely independent of your hospital's value. This means: a hospital containing $1,000,000 of stuff would still go bankrupt with $300,000 of liabilities, while one containing $50,000 could go $200,000 in debt (and technically be bankrupt with an excess of $150,000 in liabilities) and still keep running (providing it turns a profit).
A very large hospital that is running on low funds getting into a large accident that chokes its income (say, certain machines exploding creating bottlenecks in the patient flow) can find itself in $300,000 debt from a positive balance in a single month of wages and may receive an instant game-over out of nowhere.
(This part of the answer needs improvement)
I have not yet checked whether negative balance actually accrues interest. I would assume it does, and at a rate at least as high as the worst interest rate loan (so at 1.3% monthly or worse).