Savitzky–Golay filter vs. IIR or FIR linear filter
Since the discussion in the existing answers and comments has mainly focused on what Savitzky-Golay filters actually are (which was very useful), I will try to add to the existing answers by providing some information on how to actually choose a smoothing filter, which is, to my understanding, what the question is actually about.

First of all, I'd like to repeat what has become clear in the discussion spawning from the other answers: the categorization of smoothing filters in the question into linear and time-invariant (LTI) FIR/IIR filters on the one hand, and Savitzky-Golay filters on the other hand is misleading. A Savitkzy-Golay filter is just a standard FIR filter designed according to a specific criterion (local polynomial approximation). So all filters mentioned in the question are LTI filters.

The remaining question is how to choose a smoothing filter. If computational complexity and/or memory are an issue, IIR filters may be preferable over FIR filters, because they typically will achieve comparable noise suppression (i.e., stopband attenuation) with a much lower filter order than FIR filters. But note that if real-time processing is necessary, one possible disadvantage of IIR filters is that they cannot have an exactly linear phase response. So the desired signal will suffer some phase distortions. For offline-processing, the phase distortions can be avoided, even with IIR filters, by applying zero-phase filtering.

Apart from the considerations discussed in the previous paragraph, it is mainly the design criterion that matters, not so much if the filter is FIR or IIR, because any (stable) IIR filter can be approximated with arbitrary accuracy by an FIR filter, and any FIR filter can be approximated by an IIR filter, even though the latter can be much more difficult. The appropriate design criterion obviously depends on the properties of the data and the noise. When it comes to smoothing we usually assume sufficiently oversampled (i.e., smooth) data. If the noise has mainly high frequency components, i.e., if there is little spectral overlap between the data and the noise, we want to maximize stop band attenuation, or minimize stop band energy, while preserving the desired signal as well as possible. In this case we could choose a linear-phase FIR filter designed according to a minimax criterion using the Parks-McClellan algorithm. We could also minimize the stop band energy (i.e., minimize noise power in the stop band) by choosing a least squares method. A mix between the two criteria (minimax and least squares) is possible by choosing a constrained least squares design, which minimizes stop band energy while constraining the maximum approximation error in the pass band.

If the noise spectrum significantly overlaps with the signal spectrum, a more careful approach is required, and brute-force attenuation will not work well because either you leave too much noise (by choosing the cut-off frequency too high) or you distort the desired signal too much. In this case Savitzky-Golay (S-G) filters may be a good choice. The price to be paid is mediocre stopband attenuation, but one advantage is that some signal properties are preserved very well. This has to do with the fact that S-G filters have a flat pass band response, i.e.,

$$\frac{d^kH(e^{j\omega})}{d\omega^k}{\huge |}_{\omega=0}=0\quad k=1,2,\ldots,r\tag{1}$$

where $r$ is the order of the approximating polynomial and $H(e^{j\omega})$ is the filter's frequency response. Property $(1)$ guarantees that the first $r$ moments of the input signal are preserved in the output, which means that the width and height of peaks in the desired signal are well preserved.

Of course there's also a compromise between the two types of smoothing filters discussed above (high stopband attenuation and S-G). We could design an FIR filter with a certain degree of flatness at $\omega=0$ and use the remaining degrees of freedom to maximize stop band attenuation, or minimize stopband energy. In the case of FIR filters, the resulting design problem is sufficiently simple (and convex), and general optimization routines available in several software packages can be used to obtain the optimal filter for the given application.

For the ones interested in the theory of S-G filters, the most relevant references I can recommend are the following:


the original paper by Savitzky and Golay
a tutorial paper by Ronald W. Schafer
section 8.3.5 of Orfanidis' DSP book