Reduce might work better than Solve for this problem.
I started this without using any of your additional inequalities about domains and added those, one at a time, after seeing from the result that Reduce needed additional information. In the very last steps I even introduced a couple of inequalities which seem should have been deduced from what I gave it earlier, but Reduce didn't seem to be able to find those on its own. Reduce completes this calculation in a few seconds.
Reduce[{(-a+c β+t θ1)^2(6b^2+4b λ+λ^2)/(8b^3)>(a-c-t θ3)^3/4b,
0<c<a, 0<b<1, 0<λ<1, 0<t<(a-c α)/θ2, t<(a-c β)/θ1, 0<θ1<θ2<θ3<1, 0<θ3, β c+t θ1<a, α c+t θ2<a}, β]
but the result returned from that is complicated and includes lots of things which you know to be true and which clutter up what you are trying to see.
Simplify can be used to discard things which you know to be true if you include that information in the optional third argument, which in this case is just repeating your additional inequalities.
Simplify[
Reduce[{(-a+c β+t θ1)^2(6b^2+4b λ+λ^2)/(8b^3)>(a-c-t θ3)^3/4b,
0<c<a,0<b<1,0<λ<1,0<t<(a-c α)/θ2,t<(a-c β)/θ1, 0<θ1<θ2<θ3<1,0<θ3,β c+t θ1 <a,α c+t θ2<a},β],
{0<c<a,0<b<1,0<λ<1,0<t<(a-c α)/θ2,t<(a-c β)/θ1, 0<θ1<θ2<θ3<1,0<θ3,β c+t θ1 <a,α c+t θ2<a}]
which returns
β*c + t*θ1 + Sqrt[2]*b^2*Sqrt[(a - c - t*θ3)^3/(6*b^2 + 4*b*λ + λ^2)] < a ||
c + t*θ3 > a
That doesn't quite completely isolate β, but sometimes results are close enough that the extra work needed to format the result doesn't seem like it is worth the effort to me. That output also includes the alternative c + t*θ3 > a which might be eliminated with the addition of another domain inequality, but hopefully this is close enough and gives you an idea how you can do similar things in the future.
Please check all this very carefully to try to make certain I haven't made any mistakes in reformulating your problem.