Which IIR filters approximate a Gaussian filter?
The Deriche and van Vliet filters are heuristics.  In both cases they choose the locations of poles and zeros to minimize either the RMS difference or the maximum difference of the filter's impulse response from a Gaussian.

Both filters are causal-anti-causal pairs.  So I think they have no phase error or group delay, but you need to be able to run them backwards on the data as well as forwards.  This makes them popular in image processing, but perhaps restricts their applicability elsewhere.

That they are heuristics is evidenced by the wealth of papers that tweak them.  For example a google search (while I was looking for the link to the Deriche paper) turned up this one that tries to solve the problem that the Deriche derivative-of-gaussian filter doesn't have an exactly 0 DC response.  There are also some interesting issues about initializing the boundary conditions correctly.

I have found the following overview a good resource:
Dave Hale, Recursive Gaussian ﬁlters, Colorado School of Mines Center for Wave Phenomena report CWP-546.