Optimal $ n $ -th Order IIR /AR Approximation of a Moving Average Filter
(Update: I just realized that the first part of this covering CIC structures is basically what Hilmar has already answered-- I'll leave this up since it offers more graphics and details in case that his helpful to anyone but it is indeed the same answer) 

This may not be optimum and although a feedback structure is involved it is strictly not IIR, but want to point out the efficiency of the cascade integrator-comb structures for implementing boxcar moving average filters- and importantly how the results match exactly and not an approximation. So a 600 tap moving average filter can be implemented with simply one subtraction after a 601 sample delay followed by an accumulator.

These structures are quite popular when resampling is involved (often after a moving average the higher sampling rate of the input is no longer required and a lower output sampling rate would be even more efficient).



The equivalency is easily proven from the Z-transform for each of these structures. Specific to the example structures in the graphic:

The Z-transform for the accumulator is: 

$$\frac{1}{1-z^{-1}}$$

The Z-transform for the comb (delay and subtract) filter is:

$$1-z^{-4}$$

And the Z-Transform for the moving average filter is: 

$$1+z^{-1}+z^{-2}+z^{-3}$$

Combining confirms the equivalence:

$$1+z^{-1}+z^{-2}+z^{-3} = \frac{1}{1-z^{-1}}(1-z^{-4})$$
$$(1-z^{-1})(1+z^{-1}+z^{-2}+z^{-3})= (1-z^{-4})$$

Another favorite of mine as a true IIR alternative to a moving average filter is the exponentially weighted moving average filter, where assuming sufficient bit precision, very tight (in other words effectively long delay) filtering can be achieved by using an alpha that approaches 1. This is demonstrated in the following graphics: 




In fixed precision structures the order of the two operations shown in the block diagram of the exponentially weighted moving average filter is very important! In most cases when dealing with fixed precision it is best to allow digital filters to grow the signal first and do the scaling down after the accumulation. This also explains the motivation for extended precision accumulators- the accumulator shown needs to be designed with sufficient precision to prevent any overflow for the given $\alpha$ value chosen (the Infinite Summation identity $\sum_{n=0}^\infty \alpha^n =\frac{1}{1-\alpha}$ is useful in this case to determine worst case maximum value; resulting in $gain = \frac{1}{1-0.99}$ or 100 in the example shown, so in this case we would need to add $log2(100) = 7$ bits in the accumulator). Also clear from the summation identity is how the scaling term itself was determined such that the gain of the structure is 1. 
The order of these operations is no different than the reason to put an amplifier prior to an attenuator in an analog system to get the best noise figure, while paying attention to clipping and maximum input signal, for those familiar with that.