Most of this inscription (which I have translated below from Bœotian to Attic) is a list of names and contributions, but the first few lines form a sentence (repeated in the last few lines) which I am having a bit of trouble understanding.
“These [people?] contributed things for the war, which the Bœotians fought over the temple τοῦ in Delphi due to the sacrilege against the temple of the Pythian Apollo.”
The phrase τοῦ ἐν Δελφοῖς (originally τῶ ἐμ Βελφοῖς), which appears in both sentences (once reconstructed) is confusing—τοῦ doesn’t belong to a noun unless it’s repeated; it can’t seem to mean “thereof” (τῶν would make sense in this context, assuming the article can be used in that way), and it’s not being used for ὅν; moreover a war would hardly seem to be fought in Delphi (though perhaps ‘by’ falls within the semantic range of ἐν.) It would seem to suggest ‘of [those] in Delphi’, but I can’t figure out what that phrase would be modifying—could it be a way of saying “the temple owned by Delphi”?
τοῦ ἱεροῦ τοῦ ἐν Δελοῖς is a perfectly common construction. The repetition of the pronoun just joins the two parts. It’s the same temple. τοῦ ἱεροῦ τοῦ ἐν Δελοῖς = The temple in Deli. In the same way that, ὁ οἶκος ὁ καλός means “the beautiful house”, and functions as a subject (meaning that it’s not in a predicative position): ὁ οἶκος ὁ καλός ἐν τοῖς Δελφοῖς ἐστιν. The beautiful house is in Delphi. ὁ ἱερεὺς τοῦ ἱεροῦ τοῦ ἐν Δελοῖς Ἰὼν ὀνομάζεται. The priest of the temple in Deli (the priest of that temple which is in Deli) is called Ion. This is all due to the fact that Greek articles originate from the relative pronoun, and only take their known standard form consistently in Attic. Take a look at this (the 1158), it should clear the question: http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0007%3Apart%3D4%3Achapter%3D40%3Asection%3D82%3Asubsection%3D77
You should complement your mental picture of the workings of adjectives with articles with the predicative position (which is not the case here, since “τοῦ ἐν Δελοῖς” is working as an adjective equivalent to ‘Delian’). You can do that here: