The answer I give is not specific for logistic regression models, but it is also relevant for other types of regression models.
Excluding the "Don't Know's" is a very bad idea. You discard some important information. You are in some sense creating a missing-data problem. This can have more or less impacts on your results. You can create dummies for two of the categories and use the third one as a reference category. Which one you choose as a reference is mainly a matter of taste and convenience.
Likert scales can be handled in the same way. In your case, with three categories, you can create dummies for two categories and use the third category as a reference. I would not introduce it as a continuous variable. With the dummy coding you are able to capture non-linearities. This way you will obtain a much more flexible specification. More generally speaking, if you have $k$ categories, create $k-1$ dummies and use the remaining category as reference.
Depending on the software you use, the creation of dummies may be superfluous. There are ways to introduce categorical variables directly in regression models. In Stata you can have a look at
help fvvarlist and in R at
Note that there are other coding schemes you could use. Dummy coding is quite straightforward to implement and to interpret.