The phenomenon you hope for cannot occur:

If the product X × Y has the homotopy type of a finite-dimensional CW–complex, then each factor X and Y already has the homotopy type of a finite-dimensional CW–complex (in fact of dimension ≤ dim(X × Y)). Hence no counter–example exists.

1. From “finite dimensional product’’ to “finite dimensional domination’’

Assume

  h : X × Y → K and g : K → X × Y

are homotopy inverses, with K an n–dimensional CW–complex.

Choose a point y₀ ∈ Y and put

 a : X → X × Y, a(x) = (x , y₀) (inclusion)

 b : X × Y → X, b(x , y) = x (projection)

Define

 i := h ∘ a (X → K)

 r := b ∘ g (K → X).

Because g ∘ h ≃ id_{X × Y}, we obtain

 r ∘ i = b ∘ g ∘ h ∘ a ≃ b ∘ id_{X × Y} ∘ a = b ∘ a = id_X.

Thus X is dominated by K: there are maps i : X → K and r : K → X whose composite is homotopic to the identity on X.
Exactly the same construction (use a point x₀ ∈ X and the other projection) shows that Y is also dominated by K.

2. Finite-dimensional domination ⇒ finite-dimensionality

A standard result in CW–complex theory says:

 If a space Z is dominated by an n-dimensional CW–complex K,
 then Z has the homotopy type of an n-dimensional CW–complex.

Proof sketch. Replace Z by a CW–approximation (so Z is itself a CW–complex).
Because i : Z→K can be chosen cellular (cellular approximation theorem) and r∘i ≃ id_Z, Z is a homotopy retract of the n-complex K.
Attaching the mapping cylinder of i and then collapsing the copy of K gives an n-dimensional CW–complex that is homotopy equivalent to Z.
(A detailed argument appears e.g. in J. H. C. Whitehead, “Simple homotopy types”, §4, or in M. M. Cohen, “A course in simple homotopy theory”, Prop. 4.13.)

Applying this with Z = X (resp. Z = Y) and K as above shows that both X and Y are homotopy equivalent to CW–complexes of dimension ≤ n.

3. Consequence for the original question

Therefore

 (X × Y is homotopy equivalent to an n-dimensional CW–complex)
 ⇒ X and Y are each homotopy equivalent to n-dimensional CW–complexes.

In particular, there is no space X that fails to be homotopy equivalent to any finite-dimensional CW–complex yet whose product with some Y acquires that property.

So the answer to the MathOverflow question is “No; such an example does not exist.”