Normally you handle that with a "stop-loss" order that buys the stock (closes out your short) if it rises to a certain price. You're asking what happens if you don't, or can't, and it rises quickly.
They will automatically force you to buy to cover it. If that disadvantages you, they really don't care.
They will use your brokerage account's assets. Starting with cash positions obviously, but they will cheerfully sell any asset without considering the tax impact. Stocks held long obviously, shorts that are in the money, anything positive.
As for negative assets (like other shorts), as they strip assets, you have less assets to cover the broker's risk on those... so they may force their sale also. This can become a cascade.
If cashing you out isn't enough, they'll loan you the money, and you must pay it. But this means the broker really screwed up, got fantastically unlucky, or is very, very confident you have outside assets to cover it.
Then you pay, or a) you won't be doing any trading there, and b) after the mark hits your credit report, you won't be doing any trading anywhere else, either.
What if it's rising slowly? Then the broker may contact you to ask you to put more assets in the account (or voluntarily close out the short). This is called a "margin call".