Protein binds to DNA and just stays there, what will happen next?
All binding reactions are reversible. The binding and unbinding (association and dissociation) are usually in equilibrium. How much of the protein is bound to the DNA and how much time a single molecule spends in the bound state depends on the kinetic parameters. In case of the non-catalytic Cas, the protein may inhibit transcription if it is bound at the promoter. In fact, such inactive variants have been developed and used for targeted inhibition of transcription (Himeda et al., 2015).

If the CRISPR-Cas is blocking a binding site for a transcriptional activator, then the degree of inhibition would depend on the competition between the activator and the CRISPR-Cas-inhibitor for the binding sites. This competition would be dependent on the kinetic parameters as well as the concentration of the competing species. 

Similarly, you can design proteins/RNAs that can bind to mRNAs (at RBS in prokaryotes or Kozak sequences in eukaryotes) and inhibit translation. 

Another interesting fact: CRISPR-Cas system has been engineered to work like transcriptional activator, as well (Didovyk et al., 2016).