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I want to estimate my DSGE model. I first find a mode of the posterior density using csminwel (i.e. mode_compute=4)

I get the following results:

Initial value of the log posterior (or likelihood): -6278.8237

After several iterations the algorithm ends up with:

Final value of minus the log posterior (or likelihood):2995.576238

Is there any problem with this value? Should it be negative?

No, there is no problem. Why do you think so? The initial value of MINUS the posterior was 6278.8237. We are now searching for the minimum, which was found at 2995.576238

When I add some additional features in the model and run the estimation I get:

Initial value of the log posterior (or likelihood): -3165.4263
Final value of minus the log posterior (or likelihood):-119.039952

So, final value of minus the log posterior (or likelihood) can be either positive or negative number?

Yes, see http://www.dynare.org/phpBB3/viewtopic.php?f=1&t=2847