When a European option gains value with time

Theta is usually negative: options lose value as time passes. Under what circumstances can a European option have a positive theta?

When the option is deep in the money and carry dominates. A deep in-the-money European put is worth roughly $Ke^{-r\tau}-S$; as time passes the discounted strike rises, so its value increases when $r>0$. Similarly a deep in-the-money European call on a stock with a high dividend yield behaves like $Se^{-q\tau}-Ke^{-r\tau}$ and can gain value over time. Early exercise would remove this effect for American options.