Is it appropriate to put a scientific/technical document about code developed by our group on arXiv? Basically, I have written a tutorial for a code which performs a specific type of electronic-structure calculation. The code has been developed by the group I belong to and writing this tutorial/manual is part of our efforts towards achieving greater visibility and getting people to use the code.

I have put in sections on the underlying theory, some technical details, usage guidelines, and some examples of actual calculations that can be done with the code. As such, there is no original scientific contribution, since we already published the new scientific bits pertaining to the implementation in another paper. However the document is a (rigorous) scientific and technical document.

I have seen manuals uploaded on the arXiv before, but I do not know if this fits the arXiv's purpose. Hence, I was wondering whether it is appropriate/a good idea to upload my tutorial to the arXiv.

(Note: I have been browsing the arXiv help but have not found anything on this. Let me know if I have missed any policies stated somewhere on their website.)
If your manual is specific to your group's environment (software, lab, equipment, protocols, etc.), then I would consider posting such document on arXiv as not appropriate. A better place for such documents IMHO would be figshare or Zenodo. Both services allow artifacts to be not only citable, but also discoverable (via DOI assignment). Version control is supported as well, but the advantage of Zenodo in this regard in comparison with figshare is Git and GitHub integration.

If, on the other hand, the document is not specific to your group's or other particular environment (or, perhaps, you want to share this document as a model of how such documents should be structured, presented and what content should they contain), then I agree with other answers in that it is totally appropriate to post such document on arXiv.