Is it possible to measure the wave features of an elementary particle without absorbing it? If the answer to the question is no, I am wondering how we can be sure that the absorption did not alter the "actual" wave features.
 A: For a bound state you can measure the spacial distribution—in technical language the "form factor".
This is a scattering measurement, so the electrons that you are measuring influence the behavior of the beam, and it is the scattered beam that you detect.1 
The result shows that the electron's charge is distributed in space, even though direct measurements of an isolated electrons charge distribution always show as point-like to all experimentally accessible scales. That is, you are measuring the standing wave the electron(s) makes when bound to an atom.

1 I'm used to thinking of this in a nuclear physics context where we scatter an electron beam off of a nuclear target to measure nuclear form factors, but scattering x-rays off of a crystalline atomic target lets you measure the atomic form-factor (at least once you've unfolded the form-factor from the crystalline structure function). 
A: It depends what you mean by "an elementary particle". If you mean the same elementary particle, as "this specific billiard ball", the answer is no, The wave nature of a specific electron or proton or... is not measurable, because it is a quantum mechanical quantity, it appears in the predicted probability distribution, and probability distributions mean measuring many times with the same initial conditions many electrons, protons,....
This becomes clear in the double slit experiment , "electron +two slits" scattering, one electron at at time.

In frame a) one sees individual dots, each is the footprint of the elementary particle we call electron, interacting with the atoms in the film and leaving a point. No wave nature appears. Firing electrons continuously at the same slits with the same momentum builds up the probability distribution for this particular experiment, and displays the wave nature of the electron.
It is possible to see the wave nature of the generic "electron's" nature , but not that of the same electron.
A: I don't know if the type of experiment I am going to describe is what you have in mind but it seems related enough to warrant an answer. Such setups are sometimes called counterfactual or interaction-free measurements. I will shamelessly lift the introduction from the most pedagogical presentation [a] I know of. Readers with access to scientific journals can skip the following and directly go to the referenced article!
The following figure is a schematic of the experiment. The dotted (resp. solid) lines represent beam splitters (resp. mirrors). The red-ish lines represent the optical path of photons. $D_1$ and $D_2$ are two photon detectors whereas $S$ is a source which can emit one photon at a time. Finally $B$ is an removable obstacle which may block the path it sits on.

The position of the beam splitters and mirrors can be arranged so as to create destructive interference, in such a manner that $D_1$ always detect a photon whereas $D_2$ never does so. That is if $B$ is not there. If it is placed on that arm of the interferometer, then there are no destructive interferences anymore, and both $D_1$ and $D_2$ detects photon, which a probability of 1/2 each.
So let's say we fire the source $S$. There are 3 possible outcomes:


*

*neither $D_1$ nor $D_2$ detects anything

*$D_1$ detects a photon

*$D_2$ detects a photon


The first case is only possible if $B$ is in place and it has absorbed the photon. The second case says nothing: the photon may have passed through an interferometer without $B$, or it may have passed through one with $B$ but along the unblocked arm. The third case is the interesting one. It can only occur when $B$ is in place but the photon has not interacted with $B$. So basically, we detect the presence of $B$ without any particle hitting it! This is not mind boggling if you think about it: it just owes to the non-locality of the wave function.
I refer readers interested in a rigorous Quantum Mechanical treatment to [a].
[a] Avshalom C. Elitzur and Lev Vaidman, Quantum mechanical interaction-free measurements, Foundations of Physics 23 (1993), 987--997
