Let's say a State has two obligation, on one hand, it must return a person to country of origin under treaty law, on the other hand, the person seeks asylum and thus, under 1951 convention principle of "non-refoulement" applies.
does it matter that one of those obligations initiated earlier than the other one?
Is there a hierarchy among humanitarian obligation and other obligations of a State?
There are three main ways conflicts between legal effects (obligations, institutional facts, etc.) are derogated.
Lex superior - the effect originating from the highest priority legal source (e.g. human rights charters) derogates.
Lex posterior - the effect originating from the most recently enacted legislation derogates.
Lex specialis - the most specific effect derogates (e.g. any EU product can be sold in any EU country. Italy requires only pasta made from wheat can be sold as pasta. Germany makes pasta made from some other product. Italy's law is more specific and therefore German-labelled pasta which does not constitute pasta to Italy's legal system cannot be sold in Italy).
If a more specialised effect somehow contradicts an effect from a higher source, then one principle must be determined as taking precedent over another. It depends on the legal system which is which. I don't know any legal system where an effect from superior source has been derogated by an inferior source. Human Rights law should always take precedent, so should a constitution, under the lex superior principle.
Yes, there's a recognized order of obligations. In particular, states cannot bypass such pesky things as Universal Human Rights by signing a conflicting treaty.
The specific example is a case where the asylum procedure exists to protect such a universal human right, and therefore if it applies it trumps the extradition treaty.
