@Chris's [comment](https://mathoverflow.net/questions/365543/what-are-the-sections-of-an-ideal-sheaf-on-a-scheme#comment922798_365543), posted as an answer by [request](https://mathoverflow.net/questions/365543/what-are-the-sections-of-an-ideal-sheaf-on-a-scheme#comment922998_365543) (made CW to avoid reputation):

> A sufficient condition is that $U$ is affine.  Indeed $\mathcal I$ fits in an s.e.s. $0 \to \mathcal K \to \mathcal O^n \to \mathcal I \to 0$ and $H^1(U, \mathcal K) = 0$ for $U$ affine.