Consider a chain complex $C$ and a poset $P$ so that there is a filtration by subcomplexes $C^p$ of $C$ where $p\in P$ in such a way that $p<q$ implies $C^p \leqslant C^q$.


As a second option, consider the situation when $C$ is $P$-graded. Then one can set $C^p = \{ x : |x| < p\}$ and $\overline{C}^p = \{ x : |x|\leqslant p\}$. These two filtrations are related, but would give different sub-quotients. If the poset $P$ is discrete, there's no problem, however. 


Is there any literature on the possibility of producing spectral sequences out of such data, or doing homological algebra with this data? 



Poset filtrations
There is recent work on the homotopical algebra of the simplicial analoque of your situation. Lurie defined a P-stratified space as being a space over the Alexandroff space corresponding to P. Very recently in his thesis, 
<https://arxiv.org/abs/1908.01366>,
 Sylvain Douteau has looked at this from the point of view of simplicial sets over the nerve of P. This was for fixed $P$ but he then freed things up in a subsequent article on the arxiv. His abstract is:
"In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those model structures together, we construct a cofibrantly generated model structure on the category of all stratified spaces. In both model categories, weak-equivalences are characterized by stratified homotopy groups." (see <https://arxiv.org/abs/1911.04921>).


There is related material in another recent thesis:


Nand-Lal.A simplicial appraoch to stratified homotopy theory. (<https://livrepository.liverpool.ac.uk/3036209/>)


These approaches may provide ideas for handling your situation. For instance by converting the chain compolexes into simplicial abelian groups (provided the chain complexes are zero in negative dimensions) using Dold-Kan then adapting Douteau's arguments to get them to be Abelian group objects in a category of presheaves on the category of elements of $Ner(P)$. That should give a good context to study their homological algebra, e.g. by resolving by fibrant objects.


I hope this helps.


