Setting the options PlotPoints -> {2, 60}, MaxRecursion -> 0 on cylStack and washStack reduces their size by a factor of more than 10. This will speed things up.

It also help if you get rid of more global symbols (e.g hLine, vLine, washPlot, cylPlot). When they are changed inside Manipulate, the change causes a second update, which makes the responsiveness twice as bad. I used With to make them local.

Code dump:

f[x_] := 1/4 x^4;

g[x_] := 2 x;

(*Defining the function plots*)

fBound = ParametricPlot3D[{{x, 0, f[x]}, {-x, 0, f[x]}}, {x, 0, 2},

PlotStyle -> {Thick, Directive[Dashed, Gray]}];

gBound = ParametricPlot3D[{{x, 0, g[x]}, {-x, 0, g[x]}}, {x, 0, 2},

PlotStyle -> {Thick, Directive[Dashed, Gray]}];

(*Lines for Axes*)

axesLine =

Graphics3D[{{Line[{{-2, 0, 0}, {2, 0, 0}}],

Line[{{0, 0, -1}, {0, 0, 4}}]}}];

(*Fill for the region*)

fill = Graphics3D[{EdgeForm[None], Blue, Opacity[.4],

Polygon[Table[{n, 0, f[n]}, {n, 0.01, 2, .01}]]}];

(*Stacks*)washStack =

Table[RevolutionPlot3D[n, {t, 1/2 n, (4 n)^(1/4)}, Mesh -> None,

PlotPoints -> {2, 60}, MaxRecursion -> 0,

PlotStyle -> Opacity[.5],

BoundaryStyle -> RGBColor[.3, .2, .2, .5]], {n, .1, 3.9, .2}];

cylStack =

Table[RevolutionPlot3D[{n, t}, {t, f[n], g[n]},

Mesh -> None, PlotPoints -> {2, 60}, MaxRecursion -> 0,

PlotStyle -> Opacity[.4],

BoundaryStyle -> RGBColor[.3, 0, .4, .5]], {n, .1, 1.9, .1}];

Manipulate[

(*Lines for orientation*)

With[{hLine =

Graphics3D[{Red, Thick,

Line[{{1/2 wpos, 0, wpos}, {(4 wpos)^(1/4), 0, wpos}}]}],

vLine =

Graphics3D[{Red, Thick,

Line[{{cpos, 0, f[cpos]}, {cpos, 0, g[cpos]}}]}],

(*Defining the Slices*)

washPlot =

RevolutionPlot3D[

wpos, {t, 1/2 wpos, (4 wpos)^(1/4)}, {\[Theta], 0, \[Theta]max},

PlotPoints -> {2, 60}, MaxRecursion -> 0, Mesh -> None, PerformanceGoal :> "Quality"],

cylPlot =

RevolutionPlot3D[{cpos, t}, {t, f[cpos], g[cpos]}, {\[Theta],

0, \[Theta]max}, PlotPoints -> {2, 60}, MaxRecursion -> 0, Mesh -> None, PerformanceGoal :> "Quality",

PlotStyle -> Opacity[.6]]},

(*For Display*)

Show[fBound, gBound, axesLine, fill,

If[slc == 0, {hLine, washPlot}, {vLine, cylPlot}],

If[wstk, washStack, {}], If[cstk, cylStack, {}],

(*Global Properties for the SHOW Plot*)ViewPoint -> {0, -80, 12},

PlotRange -> {{-2, 2}, {-2, 2}, {-1, 4}}, SphericalRegion -> True,

Axes -> False, Boxed -> False, ViewAngle -> .013]

],

(*Manipulate Controls*){{slc, 0,

"Slice Orientation"}, {0 -> "Horizontal",

1 -> "Vertical"}}, {{wpos, 2, "Horizontal Slice Position"}, .01,

3.99}, {{cpos, 1, "Vertical Slice Position"}, .01,

1.99}, Delimiter, {{\[Theta]max, .01, "Revolution Angle"}, .01,

2 Pi}, Delimiter, {{wstk, False, "Show Washer Stack"}, {True,

False}}, {{cstk, False, "Show Shell Stack"}, {True, False}},

ControlPlacement -> Left,

TrackedSymbols -> {\[Theta]max, wpos, cpos, slc, wstk, cstk}

]