Does a measurement always require the exchange of energy?
In theory there is no lower limit on the amount of energy that must be exchanged to make a measurement, at least directly. But you are constrained by Landauer's principle when you initialize the memory you need to record the measurement. Effectively the measurement is the replication of information about the measured system's state in the measurement system's state. This means that some information about the measurement system's state - that which the measurement "displaces" - must wind up encoded in the environment and ultimately one needs to do work to expel this excess entropy.
So you will need to expend work $k\,T\,\log 2$ to initialize the memory you need to record each bit of information measured.
For some time after Szilard thought up his famous Szilard engine thought experiment (1929) he believed the mechanism whereby the engine complied with the second law was exactly that it takes an amount of work equal to $k\,T\,\log 2$ to measure one bit of information about a system's state.
However this was shown not to be so by several thought experiments, most notably the Fredkin-Toffoli Billiard Ball Computer. In this device, internal register states can be polled without expenditure of energy. An excellent review of these and other ideas is to be found in
Charles Bennett, "The Thermodynamics of Computation: A Review", Int. J. Theo. Phys., 21, No. 12, 1982
As discussed in the Bennett paper, the work input to the Szilard engine that the second law would require is needed to "forget" bits of information, as I summarize in my answer here. When you make a measurement, you need to "make room" for it by "forgetting" the former state of the system you encode that measurement in.
Addendum
Some further explanation connecting the frictionless computer with measurement. The Toffoli computer, if I understand the history correctly, was the first accepted demonstration of the error in Szilard's assumption that it was the measurement that required the amount $k\,T\,\log 2$ of energy to decide whether a molecule were travelling fro- or to-wards the door in his own version of the Maxwell Daemon. The polling of one bit of a computer's memory is exactly the same thing as the molecule measurement: it is the inference of one bit of information about an observed physical system's state and, at least in this case, the Toffoli experiment shows that this inference can be done without energy expenditure.
Now, if you are worried about friction in the Toffoli experiment, then include it in the thought experiment, and imagine decreasing it through some engineering measure: better machining, magnetic levitation, whatever. As the friction is decreased through these measures, there is no fundamental physical principle encountered which halts the process. It may be impracticable to further the process, but there is no fundamental physical reason why the friction cannot be lowered. You can in principle make it arbitrarily small. This is quite different from the realization that to infer one bit of information about a system, you have to write that information in the physical state of some system, and the state that you write over has to be recorded elsewhere. This follows from an assumption that the microscopic physical laws are reversible, and, if true, the limit Landauer limit is a fundamental one.
Yes. The Aharonov-Bohm effect is an example where something can be measured without exchanging energy.
A long solenoid inside the two paths of an electron two-slit experiment induces a phase shift $\Delta \phi=q \Phi_B / \hbar$, even if there is no external electromagnetic field. The only thing that matters is that the integrated magnetic potential around the path is nonzero [1]. So here we can measure magnetic flux without adding or removing energy to it.
I think interaction-free measurements in quantum mechanics also fits [2]. The Elitzur–Vaidman bomb-testing problem is likely the most famous example [3]. Here measurements are using counterfactual interactions, where information is gained even when there is no energy exchange (and clever setups can make it arbitrarily unlikely to happen).
[1] https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect
[2] Elitzur, A. C., & Vaidman, L. (1993). Quantum mechanical interaction-free measurements. Foundations of Physics, 23(7), 987-997. https://arxiv.org/abs/hep-th/9305002
[3] https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester
If we measure length by a scale there is no change of energy. So it depends on subjects to be measured.