Example of a space X exhibiting the Landweber non-exactness of the additive formal group over the integers? Landweber exactness gives a criterion for when a complex oriented cohomology theory $E$ can be recovered from the formal group law over $E_{*}$ determined by the complex orientation. That is it gives a criterion for the following to hold (see for example here)
$$E_{*}(X)\cong MU_{*}(X)\otimes_{MU_{*}}E_{*}$$
Since the additive formal group law over $\mathbb{Z}$ is NOT Landweber exact, when $E=H\mathbb{Z}$ this isomorphism cannot hold for all spaces $X$.
My question is: what is an example of such a space?
 A: Firstly, you ask for a space $X$.  I will instead talk about finite spectra, but they become spaces if you suspend them enough times, so that does not really make a difference.
There is a kind of tautological answer to your question as follows.  By the nilpotence technology of Hopkins, Devinatz and Smith, for suitable sequences of natural numbers $i_k$ there are generalised Moore spectra $S/(v_0^{i_0},\dotsc,v_n^{i_n})$ such that 
$$ MU_*(S/(v_0^{i_0},\dotsc,v_n^{i_n})) = MU_*/(v_0^{i_0},\dotsc,v_n^{i_n}) $$
and there are cofibre sequences
$$ \Sigma^{|v_n^{i_n}|} S/(v_0^{i_0},\dotsc,v_{n-1}^{i_{n-1}}) \xrightarrow{}  
     S/(v_0^{i_0},\dotsc,v_{n-1}^{i_{n-1}}) \to 
     S/(v_0^{i_0},\dotsc,v_n^{i_n})
$$
with the obvious effect in $MU$-homology.  Any spectrum of the form $S/(v_0^{i_0},v_1^{i_1})$ will be an answer for your question.  Examples for this $n=1$ case were already constructed by Adams before the nilpotence theory was available (and this was a big part of the motivation for the nilpotence programme).
However, one might ask for a more elementary example.  I think that one can proceed as follows.  The Hopf map $\eta\colon S^1\to S^0$ has order $2$ and so extends to give a map $\eta'\colon S^1/2\to S^0$.  Let $Q$ be the third suspension of the Spanier-Whitehead dual of the cofibre of $\eta'$.  This has cells in dimensions $0$, $1$ and $3$.  In mod $2$ homology, the bottom two cells are connected by $\text{Sq}^1$ and the top two are connected by $\text{Sq}^2$ so the cell diagram looks like a question mark and the complex is sometimes called the question mark complex.  I think it works out that $MU_*Q=MU_*/(2,v_1)x\oplus MU_*y$ with $|x|=0$ and $|y|=3$.  On the other hand, we have $H_*Q=\mathbb{Z}/2x\oplus\mathbb{Z}z$ with $x$ mapping to $x$ and $y$ mapping to $2z$, so the map $\mathbb{Z}\otimes_{MU_*}MU_*Q\to H_*Q$ is not surjective.  However, the argument is a bit intricate.  
A: Take $X=H\mathbb{F}_p$ for $p$ an odd prime. Then $MU_*(X)\cong\mathbb{F}_p[b_1,b_2,\ldots]$ where $|b_i|=2i$. In particular, the right hand side is concentrated entirely in even degrees. On the other hand, the left hand side may be computed as $\mathbb{F}_p[\xi_1,\xi_2,\ldots]\otimes E(\bar{\tau_1},\bar{\tau_2},\ldots)$, which has classes in odd degrees, e.g. $|\bar{\tau_1}|=2p-1$.
