Jim Simons’s Renaissance Technologies LLC has produced the greatest investment returns of any hedge fund. Now, it also may be facing an unusually painful tax headache.

Last week, Renaissance sent a letter to its current and former employees warning that the Internal Revenue Service could force them to pay back taxes and penalties because they invested in Renaissance’s hedge funds through the firm’s 401(k) plan and individual retirement accounts, or IRAs, without paying fees. The warning could affect other investment firms...