In literature I sometimes stumple upon the remark, that choosing priors that depend on the data itself (for example Zellners g-prior) can be criticized from a theoretical point of view. Where exactly is the problem if the prior is not chosen independent from the data?
Generally, informative priors are typically viewed as your information about parameters (or hypotheses) before seeing the data. So any data-based prior is violating the likelihood principle since evidence from the sample is coming through the likelihood function and the prior.
The $p$-values are wrong. Take a simple example. Test whether a population mean $\mu$ is equal to a particular value $\mu_0$ or not. Suppose the sample mean $\bar x$ is greater than $\mu_0$. Then it would be simply wrong to let the data guide you into testing only a one-sided alternative. Your $p$-value will be half of what it should be.
And just to be clear: The restriction $\mu \ge \mu_0$ implied by the one-sided alternative is a kind of empirical prior. (It throws away half of the possible values for $\mu$ a priori.)