Is there a statement independent from PA and does not increase the consistency strength? There are many known independence results of $\mathsf{PA}$, for example, Goodstein theorem, Paris-Harrington theorem, and the reflection principle. But these examples imply the consistency of $\mathsf{PA}$.
I guess not all independent statements of $\mathsf{PA}$ imply the consistency of $\mathsf{PA}$. I have tried to find such examples, but I can't even prove the existence of such a proposition. I would appreciate your help.

You can attack this kind of question by thinking about the Lindenbaum-Tarski algebra for PA. This can be viewed as a partial order $\preceq$ on the set of all sentences in the language of arithmetic, where we have $\phi \preceq \psi$ if and only if $PA + \psi \vdash \phi$. (It is possible to use the opposite order, but I prefer to have stronger sentences be higher in the order than weaker sentences). 
One fact about this order is that, when we use PA or other sufficiently strong theories, the order is dense: if $\phi \prec \psi$ then there is a sentence $\theta$ with $\phi \prec \theta \prec \psi$. One way to construct $\theta$ is to begin with a sentence $\chi$ which is independent of $\text{PA} + \phi + \lnot \psi$ and then let $\theta = \psi \lor (\phi \land \chi)$. 
So, to answer the question, if we let $\phi$ be a sentence provable in PA, and we let $\psi$ be Con(PA), then we have $\phi \prec \psi$, and so by density there is a sentence $\theta$ strictly between them. This sentence $\theta$ is not provable in PA, because it is strictly above $\phi$, but it does not imply Con(PA), because it is strictly below Con(PA). 
It is much more challenging to find "natural" examples of sentences independent of PA that are implied by, but do not imply, Con(PA). The general construction shows there are at least some sentences with that property, though.  The example constructed as above is essentially
$$\text{Con}(\text{PA}) \lor \text{Con}(\text{PA} + \lnot\text{Con}(\text{PA}))$$
where $\text{Con}(\cdot)$ is the Gödel/Rosser consistency sentence.
