The disjunction property in Peano Arithmetic? Let $\phi,\psi\in\Pi_1^0$ be independent of PA.
Is the disjunction $\phi\vee\psi$ independent of PA?
 A: One can in fact prove a bit more than Joel did. For example: 

Let $X$ be any r.e. set such that for any $\phi\in X$, PA$+\phi$ is consistent. Let $\psi$ be any true $\Pi^0_1$ sentence. Then there are $\Pi^0_1$ sentences $\phi_0$ and $\phi_1$ such that:
  
  
*
  
*PA $\vdash\phi_0\lor\phi_1$.
  
*PA $\vdash\phi_0\land\phi_1\to\psi$.
  
*None of $\phi_0,\phi_1,\lnot\phi_0,\lnot\phi_1$ is in $X$.
  

In particular, we can take $\psi$ to be provable and $X$ to be the set of theorems of PA, and we get a counterexample to the question.
This result is due to Per Lindström, and can be found in Chapter 7 of his beautiful and very recommended book, "Aspects of Incompleteness". The proof is an elaboration of the argument with Rosser sentences. It is a basic technical lemma used in the study of the interpretability degrees of consistent r.e. extensions of PA. (The degree of $T$ is the collection of all $T'$ that are bi-interpretable with $T$.)
For example, Lindström uses it to show that the degrees are dense, i.e., if $a\lt b$ then there is a $c$ with $a\lt c\lt b$.
By the way, the double fixed point lemma in Joel's answer is proved in Chapter 1 of Lindström's book. The reference given there is to "Theories Incomparable with Respect to Relative Interpretability", by Richard Montague, The Journal of Symbolic Logic, Vol. 27, No. 2 (Jun., 1962), pp. 195-211. As far as I can see, Smullyan does not prove this version in his "Theory of formal systems", Annals of Mathematics Studies, No. 47, Princeton University Press, Princeton, N.J. 1961. The modern presentation of Rosser sentences comes from Smullyan's book, though. The review by Kreisel in Math Reviews even called the book "the most elegant exposition of the theory of recursively enumerable (r.e.) sets in existence."
A: The answer is no, and here is a counterexample. The proof
relies on the double fixed point lemma, a generalization of
the usual Goedel fixed point lemma producing two statements
forming a fixed point with respect to a system, and I
provide a proof below. Using it, we may produce two
distinct sentences $\phi$ and $\psi$ such that


*

*$\phi$ asserts that for every proof of $\phi$, there is
a smaller proof of $\psi$, and

*$\psi$ asserts that for every proof of $\psi$, there is
a smaller proof of $\phi$.


In this case, each of these statements has complexity
$\Pi^0_1$. Let me argue that they are independent.
First, observe that both $\phi$ and $\psi$ must be true in
$\mathbb{N}$. If $\phi$ were false, then there would be a
standard proof of $\phi$, having no smaller standard proof
of $\psi$. In particular, $\phi$ would be a provable, false
statement, contradicting $\mathbb{N}\models$PA. A symmetric
argument applies to $\psi$.
Second, observe that neither is provable (meaning provable
in PA throughout). If $\phi$ were provable, then there
would be a standard proof of $\phi$, and thus there would
have to be a smaller standard proof of $\psi$, and so
$\psi$ would be true, and so there would be an even smaller
standard proof of $\phi$. Thus, there could be no smallest
proof of $\phi$, a contradiction. And the same for $\psi$.
Thus, both the sentences are true unprovable assertions,
and hence independent.
Finally, observe that the disjunction $\phi\vee\psi$ is
provable. If both $\phi$ and $\psi$ fail in a model of PA,
then that model would have proofs of both $\phi$ and
$\psi$, but neither statement could have the smallest
proof, for if it did, then the other statement would be
true, contrary to assumption. This contradicts PA, since
the smallest proof of one of them must be smaller than any
proof of the other.
Thus, we have independent $\Pi^0_1$ statements $\phi$ and
$\psi$, such that $\phi\vee\psi$ is provable.

Here is the double fixed point lemma, which I believe is
due to Smullyan, connected with his double recursion
theorem. I use $[\phi]$ here to denote the Goedel code of
$\phi$.
Double Fixed Point Lemma. Suppose that $A(x,y)$ and
$B(x,y)$ are two formulas, then there are sentences $\phi$
and $\psi$ such that


*

*$\phi$ is provably equivalent to $A([\phi],[\psi])$, and

*$\psi$ is provably equivalent to $B([\phi],[\psi])$.


Proof. Let $\text{Sub}$ be the substitution operator, the
primitive recursive function such that
$\text{Sub}([\eta(x,y)],n,m)=[\eta(n,m)]$. Let
$\theta_1(x,y)=A(\text{Sub}(x,x,y),\text{Sub}(y,x,y))$ and
$\theta_2(x,y)=B(\text{Sub}(x,x,y),\text{Sub}(y,x,y))$. Let
$n=[\theta_1(x,y)]$ and $m=[\theta_2(x,y)]$. Finally, let
$\phi=\theta_1(n,m)$ and $\psi=\theta_2(n,m)$.
Observe that $\phi\iff \theta_1(n,m)\iff
A(\text{Sub}(n,n,m),\text{Sub}(m,n,m))$ $\iff
A([\theta_1(n,m)],[\theta_2(n,m)])\iff A([\phi],[\psi])$.
Also observe $\psi\iff \theta_2(n,m)\iff
B(\text{Sub}(n,n,m),\text{Sub}(m,n,m))$ $\iff
B([\theta_1(n,m)],[\theta_2(n,m)])\iff B([\phi],[\psi])$,
as desired. QED
Note that we can arrange that $\phi$ and $\psi$ are
distinct simply by ensuring that $\theta_1(n,m)$ and
$\theta_2(n,m)$ are not syntactically the same sentence,
such as by replacing $\theta_1(x,y)$ with its conjunction,
but ensuring that $\theta_2(x,y)$ does not have such a
form.
The lemma easily generalizes to any size system and indeed,
to infinite systems of fixed points.
A: I think Carl's answer has the same error I originally made--it doesn't address that both formulas have to be $\Pi_1^0$.  The negation of a $\Pi_1^0$ formula is $\Sigma_1^0$.
I think the actual answer is "yes" but several edit attempts haven't got the proof right, so I'll see if I can fix it offline instead of keeping on repeatedly editing.
A: This is not actually an answer but rather a comment to Joel's answer.
I am not very good in models, so here is an idea how to do without them.
There is a theorem of Kreisel: if a $\Pi_1^0$ statement is provable
in $T+\neg Con(T)$, then it is provable in $T$. In $PA+\neg Con(PA)$
we may prove that there exists the smallest code of a proof of $\phi$
and the smallest code of a proof of $\psi$. Denote them by $n_{\phi},n_{\psi}$.
Then $\phi$ asserts that $n_{\psi}<n_{\phi}$ while 
$\psi$ asserts that $n_{\phi}<n_{\psi}$. Then $\phi\vee\psi$
means $n_{\phi}\neq n_{\psi}$ which is provalbe if $\phi$ and $\psi$
are syntactically different. By Kreisel, $\phi\vee\psi$ is provalbe 
in $PA$. (Note that the numbers $n_{\phi},n_{\psi}$ do not really exist).
