Okay... let's see if word origins have anything to do with the "i before e" rule...
Weird: Old English wyrd ‘destiny,’ of Germanic origin
Ancient: late Middle English: from Old French ancien, based on Latin ante ‘before.’
Neither: Middle English: alteration (by association with either) of Old English nawther
Deity: Middle English (denoting the divine nature of God): from Old French deite, from ecclesiastical Latin deitas (translating Greek theotēs ), from deus ‘god.’
Rein: Middle English: from Old French rene, based on Latin retinere ‘retain.’
Now I'm gonna try a few of my own...
Reign: Middle English: from Old French reignier ‘to reign,’ reigne ‘kingdom,’ from Latin regnum, related to rex, reg- ‘king.’
Weigh: Old English wegan, of Germanic origin; related to wagon and wain, and to Dutch wegen ‘weigh,’ German bewegen‘move,’ from an Indo-European root shared by Latin vehere ‘convey.’
Now some that follow the rule...
Receipt: late Middle English: from Anglo-Norman French receite, from medieval Latin recepta ‘received,’ feminine past participle of Latin recipere . The -p- was inserted in imitation of the Latin spelling.
Believe: late Old English belȳfan, belēfan, alteration of gelēfan, of Germanic origin; related to Dutch geloven and German glauben, also to lief.
Beige: mid 19th century (denoting a usually undyed and unbleached woolen fabric of this color): from French, of unknown ultimate origin.
Neighbor: Old English nēahgebūr, from nēah ‘nigh, near’ + gebūr ‘inhabitant, peasant, farmer’ (compare with boor)
Friend: Old English frēond, of Germanic origin; related to Dutch vriend and German Freund, from an Indo-European root meaning ‘to love,’ shared by free.
Conclusion? So far, none. This might take further studies and analysis... maybe... doubt it... :-/