I do not have specific experience with Dirichlet distribution, but here the basic issue seems to be: A Dirichlet distribution with 4 parameters lives in a 3D space, i.e. you need to use 3 variables. So - you can rather try e.g.:

Table[DensityPlot3D[PDF[DirichletDistribution[{\[Alpha]1, 3, 2, 4}], {x, y, z}], {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, PlotRange -> All, PlotPoints -> 30], {\[Alpha]1, {1, 2, 3}}]

or

Table[SliceContourPlot3D[PDF[DirichletDistribution[{\[Alpha]1, 3, 2, 4}], {x, y, z}], {{"ZStackedPlanes", 5}, {"YStackedPlanes", 5}}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, PlotRange -> All, 
  PlotPoints -> 30], {\[Alpha]1, {1, 2, 3}}]

Does that make sense to you?