Plotting a Piecewise function that returns implicit equations using ContourPlot

Try these two changes. First, use Boole instead of Piecewise, maybe like this

pw[p1_, p2_] :=  With[{b = Boole[p1 < p2]}, 
    (p1 - 20 - 10 b)^2 + (5 p2 - 40 - (3 p2 + 20) b)^2 == 300 - 50 b
]

Second, use pw[p1,p2] (without the underscores) in the ContourPlot command.

ContourPlot works on an equation u == w by comparing values of u and w (or possibly by comparing u - w and 0, but it is equivalent). Further it relies on the continuity of u and w for its recursive subdivision algorithm and interpolates their numerical values at the mesh points to construct the linear segments of the contour, but that's not really the issue here; however it points to why ContourPlot needs access to the numerical values of the two sides of the equation. Your pw is not of the form u == w (full form Equal[u, w]). It has the equations buried inside it and returns only a boolean value. ContourPlot does not look any deeper into the expression. The fix is to bring the Equal outside the Piecewise. Here is one way to do that, by replacing the Equal in a[] and b[] by Subtract.

a[p1_, p2_] := (p1 - 30)^2 + (2 p2 - 60)^2 == 250;
b[p1_, p2_] := (p1 - 20)^2 + (5 p2 - 40)^2 == 300;
pw[p1_, p2_] := Piecewise[
   {{Subtract @@ a[p1, p2], p1 < p2}}, 
   Subtract @@ b[p1, p2]]  == 0;
Grid[{{
   ContourPlot[Evaluate@a[p1, p2],  {p1, 0, 50}, {p2, 0, 50}], 
   ContourPlot[Evaluate@b[p1, p2],  {p1, 0, 50}, {p2, 0, 50}], 
   ContourPlot[Evaluate@pw[p1, p2], {p1, 0, 50}, {p2, 0, 50}]}}]