Applications forget setting in every boot I have an android tablet with ICS version. The model is not popular. The brand is something like a Greek bestbuy.com shop. But the hardware is enough good. Nvidia tegra 3, good resolution and ram. 

My problem is that when I don't use the tablet I shut it down. But then I reopen it some apps don't remember my login credentials or other settings. These apps is facebook, Microsoft OneNote, Tasks for MS exchange, Skydrive or Skype.

What kind of problem is that? I mean is a application bug, android or model bug?
It could be a manufacturing fault in your particular tablet, but it sounds more likely to be an error in that model. Perhaps the system integrator didn't put applications' data on non-volatile storage, or screwed up the shutdown sequence. If this is the case, updating the system may fix the problem. Check with the manufacturer.

If you edit your question to include the make model name of the tablet, you may find someone else who has one and can help.