The UnitStep discontinuities are assumed to be discontinuities in the entire expression. It's not easy to get rid of all the spurious ones in general. The following seems to work in this case:
d = 1; c = 1/4; \[Alpha] = 3/5; \[Lambda] = 1/2; z$lower = .2;
z$upper = .5; y$upper = 1;
F$G8 = Plot3D[{-((c - d)^2/(2 (-2 + \[Delta] + 2 \[Lambda])))
UnitStep[
c/\[Delta] - (c - d)/(
2 (-2 + \[Delta] + 2 \[Lambda]))] UnitStep[
6 - (UnitStep[((
2 (c - d) \[Beta])/((-2 + \[Delta] +
2 \[Lambda]) (-2 + 8 \[Beta] + \[Delta] +
2 \[Lambda]))) + ((
2 (c - d) \[Beta])/((-2 + \[Delta] +
2 \[Lambda]) (-2 + 8 \[Beta] + \[Delta] +
2 \[Lambda]))) - ((0) + ((-c + d)/(-2 +
8 \[Beta] + \[Delta] + 2 \[Lambda])))] +
UnitStep[(
2 (c - d) \[Beta])/((-2 + \[Delta] + 2 \[Lambda]) (-2 +
8 \[Beta] + \[Delta] + 2 \[Lambda]))] +
UnitStep[
c/\[Delta] - (
2 (c - d) \[Beta])/((-2 + \[Delta] + 2 \[Lambda]) (-2 +
8 \[Beta] + \[Delta] + 2 \[Lambda]))] +
UnitStep[-(-2 + \[Delta]^2/(2 - 2 \[Lambda]) +
2 \[Lambda])] +
UnitStep[-(1/
8 (-2 - 8 \[Beta] + \[Delta] + 2 \[Lambda]))] +
UnitStep[-(((c - d)^2 \[Beta])/(
2 (-2 + \[Delta] + 2 \[Lambda]) (-2 +
8 \[Beta] + \[Delta] +
2 \[Lambda]))) - (-(((c - d)^2 (-2 +
2 \[Beta] + \[Delta] + 2 \[Lambda]))/(
4 (-2 + \[Delta] + 2 \[Lambda])^2)))]) - 1/10000]} //
PiecewiseExpand[#, 0 < \[Delta] < 1/2 && 0 < \[Beta] < 1/2,
Method -> {"ConditionSimplifier" -> (Reduce[#, \[Delta]] &)}] & //
FullSimplify[#, 0 < \[Delta] < 1/2 && 0 < \[Beta] < 1/2] & //
Evaluate, {\[Delta], 0, .5}, {\[Beta], 0, .5},
AxesLabel -> {"\[Delta]", "\[Beta]"},
PlotPoints -> 30,(*MaxRecursion->5,*)
PlotRange -> {{0, .5}, {0, .5}, {z$lower, z$upper}},
PlotStyle -> Cyan, ExclusionsStyle -> None]
