Chow's lemma for non reduced schemes? Recall Chow's lemma: 

Chow's Lemma: If $X$ is a scheme that is proper over a noetherian base $S$ then
  there exists a projective $S$-scheme $X'$ and a surjective
  $S$-morphism $f : X'\to X$ that induces an isomorphism $f^{{-1}}(U)
 \simeq U$ for some dense open $ U\subset X$.

I just looked at the proof on wikipedia (which presumably the one from EGA II) and found a slightly bothering detail. It starts by reducing to the irreducible case with the following argument:

Argument: Let $U_i \subset X_i$ be the open dense subsets in each irreducible component which satisfy the lemma. Then $U := \coprod_i (U_i -\bigcup_{j \ne i} X_j)$ is open dense in $X$ and satisfies the lemma. 

If we interpret dense as dense in the zariski topology then everything works fine throughout. But if we want it to mean scheme theoretically dense there's a problem with the argument as their might be an embedded point in $U_i \cap X_j$ (consider two lines intersecting at a double point).
Is there a way to "fix" the argument so that Chow's lemma will be true for dense = scheme theoretically dense? 
If not is there a simple example of a non-reduced scheme for which this "strong" Chow's lemma fails?
 A: It is time to carefully put together what was said in the comments and to fill in the gaps. We will show that the 'strong' version of Chow's Lemma does not hold. More precisely: there exists $X$ such that any morphism $f\colon X'\to X$ from a projective scheme $X'$ fails to be an isomorphism over a scheme-theoretically dense open of $X$, even for proper irreducible (non-reduced, of course) schemes over an algebraically closed field. Of course, there will still be a Zariski-dense open. The good news is, inspection of the Stacks Project's proof (tag 0200) shows that the strong version holds as soon as $X$ has enough scheme-theoretically dense affine opens to constitute an open cover.
We work over some algebraically closed field $k$.
Observation: To get a counter-example, it suffices to construct a proper $k$-scheme which does not admit an affine scheme-theoretically dense open.
In fact, if a proper $k$-scheme $X$ satisfies the strong lemma of Chow, i.e., if there exists a morphism $f\colon X'\to X$ where $X'$ is projective and which is an isomorphism over a scheme-theoretically dense open $U\subset X$, then $U\cong f^{-1}U\subset X'$ is quasi-projective and contains all the associated points of $X$.
By Stacks' tag 01ZY, the finite subset $\mathrm{Ass}(X) =: T\subset U$ is contained in an affine open $V\subset U$; any such $V$ is then a scheme-theoretically dense affine open in $X$.
The strategy is to start with a proper variety $X$ which contains a finite subset $T\subset X$ of points which is not contained in any affine open and then pass to another proper scheme $Y$ where the points of $T$ are thickened to become associated points.
Let me now explain one way to accomplish the thickening. (You can jump to the lemma below if you don't care how it's done.) The approach indicated in the comments also seems to work, after appropriate modifications.
Assume that $X$ is a locally noetherian scheme and let $T\subset X$ be a closed sub-scheme with ideal sheaf $\mathscr{I}\subset \mathcal{O}_X$.
Consider the sub-scheme $Y\subset \mathbb{A}^1_{X} = \mathrm{Spec}(\mathcal{O}_X[t])$ defined by the ideal $\mathscr{J} := t^2+t\mathscr{I}$ with its natural morphism $\varphi\colon Y = \mathrm{Spec}(\mathcal{O}_X[t]/\mathscr{J})\to X$. (I hope everyone can deal with this abuse of notation?) The projection $\mathcal{O}_X[t]\to \mathcal{O}_X$ onto degree zero descends to $\mathcal{O}_Y$ and this gives rise to a section $\iota\colon X\to Y$ of $\varphi\colon Y\to X$; thus, $X$ is in a natural way a sub-scheme of $Y$ and it is easy to see that if $X$ is reduced, this gives rise to a canonical isomorphism $X\cong Y_{red.}\subset Y$. More importantly, in any case, it shows that $\mathcal{O}_Y = \mathcal{O}_X\oplus t\mathcal{O}_X$ and $t\mathcal{O}_Y = t\mathcal{O}_X$.
In particular, $$\mathrm{ann}_{\mathcal{O}_Y}(t) = \mathscr{I}+t\mathcal{O}_Y = \mathscr{I}+t\mathcal{O}_X$$ and, therefore, if $\eta\in X$ is the generic point of an irreducible component of $T$, so that $\mathscr{I}_\eta\supset\mathfrak{m}_{X,\eta}^n$ for some $n$, then $\mathrm{ann}_{\mathcal{O}_Y}(t)_\eta = \mathscr{I}_\eta+t\mathcal{O}_{X,\eta}\supset \mathfrak{m}_{X,\eta}^n+t\mathcal{O}_{X,\eta}\supset \mathfrak{m}_{Y,\eta}^n$. Thus, $\mathfrak{m}_{Y,\eta}$ is an associated prime of $\mathcal{O}_{Y,\eta}$.
This shows:
Lemma — Let $X$ be a locally noetherian scheme and $T\subset X$ a closed sub-scheme. Then there exists a bijective closed immersion $\iota\colon X\to Y$ which is an isomorphism outside $T$, and which maps the generic points of $T$ to associated points of $Y$. Furthermore, $\iota$ has a retraction $Y\to X$ which is a finite morphism.
Finally, let $X$ be a proper, integral $k$-variety with a finite set of closed points $T\subset X$ which is not contained in an affine open of $X$. For example, we can take $X$ to be any non-projective, proper, normal, integral surface and $T$ the set of non-factorial points (cf. Kleiman, Toward a numerical criterion of ampleness, IV Cor. 4). Alternatively, as Takumi Murayama has pointed out in the comments, we could take Hironaka's non-projective three-fold and two points, one on each of the two curves involved in the construction except the intersection points. 
Let $Y\supset X$ be the thickening of $T$ as described in the lemma. By finiteness of the morphism $Y\to X$, $Y$ is proper too.
Then $Y$ admits no scheme-theoretically dense affine open since $\iota\colon X\to Y$ is affine and so any such would give rise to an affine open in $X$ containing all of $T$.
By the above observation, this is all we need to get a counter-example.
