What are the potential dangers (if any) facing the twin girls recently born in China with their CCR5 gene modified?
Important notes:
  
  I am not going into the ethical aspects of editing/removing CCR5 in human embryos, neither will I discuss potential effects of introducing that mutation into the human population. Both of these are very important issues, but out of the scope for this answer.
  As of now I'm not aware of any reliable sources of what was actually done in this case, so I will also keep my answer general / hypothetical without going into any specifics of these particular girls.


That out of the way lets get into the potential problems that could be caused by removing the CCR5 gene from  human embryo using the CRISPR-cas9 system. These can mainly be grouped into two classes: off-target or side effects of the procedure itself and any effects of a non-existing/functional CCR5 by itself.
One additional point I want to mention is that removal of CCR5 does not make one completely immune to HIV, because some strains of the virus (can) use the CXCR4 as a co-receptor instead.

1. CRISPR-Cas9 off-target effects  

The CRISPR-Cas9 system has become very popular with scientists in the last few years, because it allows genome editing in very many organism with previously unmatched efficiency and speed. However, people only recently  - when the potential medical use of the system was being considered - started to look into the specificity of the system. What they found was, that while CRISPR-Cas9 is supposed to be very sequence specific it often also induces mutations in genes that were not supposed to be targeted.
There is a quite a lot of ongoing research to reduce these off-target effects, one good example being to use two variant copies of the system that each only cut on strand of the DNA at neighbouring sites (instead of cutting both strands at one site as is normally done).
If such mutations occur while genome editing an embryo the possible effects are very hard to predict - it could be anything from completely harmless to cancer inducing. If one wants any reliable estimate for this, it would be necessary to check for any off-target mutations by sequencing the whole genome of the edited embryo. That however requires (even with very modern techniques) removing at least a single cell from the growing embryo before the IVF procedure can take place and might therefore reduce chances of the IVF being successful.

Main source: Extensive review to CRISPR off-target effects

2. Side-effects of CCR5 removal

Normally it would be very hard (read: basically impossible) to predict what potential side effects the removal of a specific gene would have, even if there are e.g. mice where this has already been done but only studied for specific medical aspects. In this particular case however, we are in luck because there is a naturally occurring mutation called CCR5-Δ32 that also renders the protein completely dysfunctional. We can therefore pretty safely assume that the CCR5 removal by CRISPR-Cas9 will have similar effects to this mutation.  

Since CCR5 is a gene expressed by immune cells, it  - unsurprisingly - has influences on various immune functions, but mostly seems to be affect infectious diseases. As this review puts it:


  In summary, there are ample data to suggest that CCR5 is involved in both positive and negative regulation of immune functions that relate to the regulation of leukocyte trafficking during infectious and postinfectious diseases. 


One particular risk that seems to correlate with non-functional CCR5 is both initial infection and disease progression by flaviviruses. It has been reported that the CCR5-Δ32 mutation increases the risk of being affected by both the Tickborne encephalitis (TBE) virus and West Nile Virus, both of which can affect the nervous system if the diseases progresses far.