No, in general E(XY) < E(X'Y') is not true.

First, if the variables are not independent, here is a counter example:

X = 0, 1 with equal probability

Y = X (not independent of X)

X' = 0, 100 with equal probability

Y' = 100 - X' (not independent of X')

E(X) = E(Y) = 0.5

E(X') = E(Y') = 50

E(XY) = 0.5

E(X'Y') = 0

Second if they were independent E(XY) = E(X)E(Y), and E(X'Y') = E(X')E(Y')

The expectations still have to be positive.

Example:

E(X) = E(Y) = -1

E(X') = E(Y') = 0

E(XY) = E(X)E(Y) = 1

E(X'Y') = E(X')E(Y') = 0