You're right, I didn't understand, I read F(r) as r being scalar r rather than vector r. The definition of Curl does not include any time-derivatives, thus a field is considered conservative at any given instant. So you can have F(r,t) be conservative for all t. If you want to think of a conservative field as have a zero integral of F.dr for any path, and this definition applies for instantaneous paths.
If you want to consider a finite-time path r(t) with F(r,t), this doesn't apply to the definition of conservative, and in fact you can always get a non-zero integral of F.dr. For example, start at a point at time t, go in dx direction while F has one value, then when F has a different value at another time t, go backwards on the same path. You have a closed path but non-zero integral of F.dr.
I hope I understood your question this time, and that my answer makes sense...[/b]