In vibrational spectroscopy, what affect do overtones have on the spectra?
Since a spectrum, including its overtones, is essentially a property of a molecule, one cannot really say how the overtones affect a spectrum. So, I will answer this question by first describing what an overtone is, giving an example of an effect that overtones sometimes participate in, and then an example of how overtones can be useful.

Overtones are charatceristic vibrational frequencies of a molecule which take place at approximately integer multiples of the $v=0$ vibrational modes. The existence of these overtones demonstrate that molecular vibrations are anharmonic in nature. This follows from the fact that for a purely harmonic vibration, the only non-zero transitions are those for which $\Delta v=\pm 1$. That is, harmonic oscillators can only move on state at a time. Thus, the fact that we observe these overtones tell us that the vibrations are not harmonic or anharmonic.

Now, in IR and Raman spectroscopy, there is an effect called Fermi Resonance. This effect basically allows for the mixing of multiple vibrational states. If I'm not mistaken, this also only happens for anharmonic vibrations because energy needs to be allowed to be shared between the two states which is not possible if the vibrations are pure harmonic, normal modes. I bring this up because it is quite common that one observes a Fermi resonance between an overtone and a higher frequency ground state mode. To quote the Wikipedia page,


  High resolution IR spectra of most ketones reveal that the "carbonyl band" is split into a doublet. The peak separation is usually only a few cm$^{−1}$. This splitting arises from the mixing of νCO and the overtone of HCH bending modes.


Thus, if you wish, one effect that overtones have on IR spectra is their ability to participate in Fermi resonance, which leaves the appearance of a peak being split.

Sometimes it can be difficult to know if a peak being observed is in fact an overtone. Generally, overtones have low intensity, but if they participate in a Fermi resonance, they gain intensity and will look like a normal ground state vibration (depending on what mode they come from). 

One place I have seen something likely similar to this is in the following paper[1]. They compute the IR spectrum of a water cluster, including the bending overtone, and show that it is likely the experimental spectrum was mis-assigned. In this example, being able to compute a spectrum independent of the bending overtone is quite useful.

Experimentally, overtones can be quite useful if there is a large number of peaks around the same frequency as a particular mode you are interested in studying. For instance, it is thought that the water dimer may play a significant role in absorption of radiation in the atmosphere and thus contribute to global warming. It is very hard, however, to determine the abundance of water dimers in the atmosphere because of the large amount of water vapor which covers up its entire infrared spectrum. Thus, ref. [2] used the fourth overtone of the $\ce{O-H}$ stretch to experimentally measure water dimers in the atmosphere.

Another example of the same use of overtones is in ref. [3]. There, the authors are interested in studying a possible intramolecular hydrogen bond, but one cannot just look at the lowest energy vibrations because they are swamped by a regular $\ce{O-H}$ stretch. Thus, the authors record infrared spectra in the regime where one would expect to find the first, second, third, and fourth overtones. It is confirmed that these are the peaks observed by simply computing the spectra from first principles. Thus, one also learns which geometric configurations cause which overtone peak.

In summary, overtones are not so much something that "affect" a spectrum, but are a necessary part of the spectrum because we live in a world where molecules vibrate anharmonically. I have also given examples of when overtones are quite a useful tool.

Just to get you thinking a bit, if and when you learn about two-photon or two-dimensional spectroscopies, imagine how overtones can then be used to gain even deeper insight into the molecules of interest.



[1]: Wang, Y., &amp; Bowman, J. M. (2013). IR spectra of the water hexamer: Theory, with inclusion of the monomer bend overtone, and experiment are in agreement. The journal of physical chemistry letters, 4(7), 1104-1108.

[2]: Pfeilsticker, K., Lotter, A., Peters, C., &amp; Bösch, H. (2003). Atmospheric detection of water dimers via near-infrared absorption. Science, 300(5628), 2078-2080.

[3]: Howard, D. L., Jørgensen, P., &amp; Kjaergaard, H. G. (2005). Weak Intramolecular Interactions in Ethylene Glycol Identified by Vapor Phase OH− Stretching Overtone Spectroscopy. Journal of the American Chemical Society, 127(48), 17096-17103.