Can a infinite time Turing machine perform hyper-computation like checking the consistency of the set theory ZF without using infinite memory?
I think we have a reference here to Joel Hamkin's model of infinite time computation, not just some made up idea of infinite time machines. In that model time is measured by ordinal numbers. The machine has access to an infinite tape. After $\omega$ steps, every cell of the tape has been potentially written to, and we can't throw away any cell because its content may be needed during further computation. So yes, you'd need infinite space, but that's not a problem when you've got infinite time already.