Hyperbolic Voronoi diagram for the Poincaré model, using RegionPlot

EDIT

(incorporating comments by J. M.: DistanceFunction -> dis and pre-computation of nearest function):

This is not efficient. Just rewriting metric (apologies for errors). In the following I used ContourPlot but DensityPlot could be used.

dis[a_, b_] := Abs[ArcCosh[1 + 2 ( a - b).(a - b)/((1 - a.a) (1 - b.b))]]
vh[n_] := Module[{p = RandomPoint[Disk[], n], nf},
nf = Nearest[p, DistanceFunction -> dis];
ContourPlot[First[nf[{x, y}]], {x, -1, 1}, {y, -1, 1}, 
RegionFunction -> Function[{u, v}, u^2 + v^2 <= 1], 
Epilog -> {Red, PointSize[0.02], Point[p]}, PlotPoints -> 50]]

vh visualizes using Nearest with dis as distance function.

Some examples: Range[5, 45, 5] (takes quite some time):

Apologies for errors (typographical and conceptual). I look forward to much better answers.

Here is some code I have for making fake Voronoi diagrams, adapted to the Poincaré disk model. The result has the look and feel of having been drawn with a charcoal pencil, which may or may not be desired for your application. The strategy is adapted from suggestions by Worley and Schlick.

(* some points *)
BlockRandom[SeedRandom[42, Method -> "MersenneTwister"]; 
            pts = RandomPoint[Disk[], 35]];

poincareMetric[u_?VectorQ, v_?VectorQ] := 
               Abs[ArcCosh[1 + 2 SquaredEuclideanDistance[u, v]/((1 - u.u) (1 - v.v))]]

(* Schlick's "bias" function, following Perlin and Hoffert *)
bias[a_, t_] := t/((1/a - 2) (1 - t) + 1)

With[{nodeFun = Nearest[pts, DistanceFunction -> poincareMetric]},
     Quiet @ DensityPlot[bias[0.99, HarmonicMean[#] - First[#]] & @
                         Map[poincareMetric[{x, y}, #] &, Take[nodeFun[{x, y}, 2], 2]],
                         {x, y} ∈ Disk[], AspectRatio -> Automatic, 
                         ColorFunction -> GrayLevel, Epilog -> {Thick, Circle[]}, 
                         PlotPoints -> 150, PlotRange -> All]]

The first argument of the bias[] function can be adjusted as seen fit. The following image is the result of setting the first parameter to 0.9:

where the dots corresponding to the original point positions become more pronounced, at the expense of darkening the shading within the cells.

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For completeness, here is the result of using the Beltrami-Klein metric instead:

beltramiMetric[u_?VectorQ, v_?VectorQ] :=
        Abs[ArcCosh[(1 - u.v)/Sqrt[(1 - u.u) (1 - v.v)]]]

With[{nodeFun = Nearest[pts, DistanceFunction -> beltramiMetric]},
     Quiet @ DensityPlot[bias[0.99, HarmonicMean[#] - First[#]] & @
                         Map[beltramiMetric[{x, y}, #] &, Take[nodeFun[{x, y}, 2], 2]],
                         {x, y} ∈ Disk[], AspectRatio -> Automatic, 
                         ColorFunction -> GrayLevel, Epilog -> {Thick, Circle[]}, 
                         PlotPoints -> 150, PlotRange -> All]]

Here is another approach for generating fake Voronoi diagrams. This also uses Nearest[] with the Poincaré disk metric, but uses Quílez's gradient normalization (similar to the approach used in this answer).

With the same definition for pts and poincareMetric[] as in my other answer:

poincareGradient[u_?VectorQ, v_?VectorQ] := 
        4 Sinh[poincareMetric[u, v]/2]^2 (u/(1 - Norm[u]^2) +
        (u - v)/SquaredEuclideanDistance[u, v])/Sinh[poincareMetric[u, v]]

smoothStep = Compile[{{a, _Real}, {b, _Real}, {x, _Real}}, 
                     Module[{t = Min[Max[0, (x - a)/(b - a)], 1]}, 
                            t t t ((6 t - 15) t + 10)], RuntimeAttributes -> {Listable}];

DensityPlot[With[{dm = Take[nf[{x, y}, 2], 2]}, 
                 smoothStep[0.01, 0.005, (poincareMetric[{x, y}, dm[[2]]] -
                                          poincareMetric[{x, y}, dm[[1]]])/
                                         Norm[poincareGradient[{x, y}, dm[[2]]] - 
                                              poincareGradient[{x, y}, dm[[1]]]]]],
            {x, y} ∈ Disk[{0, 0}, 1 - Sqrt[$MachineEpsilon]], AspectRatio -> Automatic, 
            ColorFunction -> ColorData[{"GrayTones", "Reverse"}],
            Epilog -> {Thick, Circle[]}, PlotPoints -> 75, PlotRange -> All]

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A similar technique can also be used to fake Voronoi diagrams on the Poincaré ball (with a quarter of the ball cut away to reveal internal structure):

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This approach is of course still usable if you want to use the Beltrami-Klein metric instead; one only needs to derive the gradient expression for the Beltrami-Klein metric and then you can proceed analogously.