It's not that there is an expiration date, but instead that there is a collateral requirement for holding short positions in addition to the interest. As the shorted stock increases in value, the borrower must post additional collateral or close out their position. If the collateral requirements exceed the fund's capital then they'll be forced to close.
Further, all hedge funds have internal risk limits, which include limits of their exposure to a single equity. Should the collateral requirements exceed this limit, then their internal risk compliance team will ensure that the short position be at least partially closed to keep them within their limits.