Why aren't some words like "Maranatha" or "Talitha Cumi" translated in English Bibles? Why aren't certain words like Maranatha, Talitha Cumi, Eli Eli Lama Sabachthani, etc., not translated in the Bible? Rather, the words are directly written and then their meaning or interpretation?

As the whole Bible is translated from some other languages, why leave certain words alone untranslated?
There are specific reasons that can be identified in each instance, but the three you mention share one important feature: they are Semitic (i.e. Aramaic or Hebrew) words amid the (otherwise) Greek New Testament. The most basic reason for an English translation to transliterate1  rather than translate these terms, then, is to reproduce what a Greek reader would have seen: a text in a familiar language with a foreign word transliterated into the alphabet of the surrounding text. 

To address each of the OP's examples and their particularities: 


Maranatha is from 1 Cor 16:22. While there is uncertainty about the precise meaning2, it is clearly an Aramaic phrase pertaining to the return of Christ. 


  If any man love not the Lord Jesus Christ, let him be Anathema Maranatha. (KJV)


The disadvantage of such a translation is that it requires a footnote for most English readers to follow. Many modern versions accordingly translate, as the ESV:


  If anyone has no love for the Lord, let him be accursed. Our Lord, come!


This loses the "preservation of foreignness feel" mentioned above, but the translators have prioritized clarity of meaning in this case.

In both of OP's other examples, modern translations generally do transliterate, for overlapping reasons:
Mark 5:41


  Taking her by the hand he said to her, "Talitha cumi," which means, "Little girl, I say to you, arise." (ESV)


Talitha cumi (again an Aramaic phrase transliterated in the Greek text) is the only real option if one wants to preserve a formal translation of the verse, since the author already provides a translation. If it were instead translated, we would have the rather silly rendering:


  Taking her by the hand he said to her, "Little girl, I say to you, arise," which means, "Little girl, I say to you, arise."

Your final example is the most complicated.3  You have quoted from Matthew 27:46:


  And about the ninth hour Jesus cried out with a loud voice, saying, “Eli, Eli, lema sabachthani?” that is, "My God, my God, why have you forsaken me?" (ESV)


As in the prior example, the author provides a Greek translation, so it would be redundant if the transliteration were not preserved. In this case we can additionally identify the author's likely motivation for doing so: in the following verse the bystanders conclude


  'This man is calling Elijah.'


Here the Hebrew "Eli" ("my God", phonetically similar to a diminutive form of the name Elijah) was needed in order for the Greek reader to follow the bystanders' confusion. This applies equally to a reader of an English translation. 




Notes: 


Transliteration is the process of representing the sounds of one language in the alphabet of another. Since Hebrew/Aramaic, Greek, and Latin/English have three different alphabets, transliteration occurs at every stage in these cases. 
The question is around how to divide the Aramaic words, which may either be construed as indicative ("Our Lord has come") or imperative ("Our Lord, come!").
I have outlined some of the difficulties with this cross-linguistic puzzle in a question on Hermeneutics.SE.