I'm not sure that "thanks be to God" is actually a translation of a dative of possession given that "gratias agere" is more an idiomatic verb meaning "to thank" which simply takes a dative object. The 'literal' translation of the phrase "gratias deo ago" would probably be more like "I direct my thanks to you" in which case the dative just plays it's normal role of answering "to/for whom?". The phrase "thanks be to God" is probably just an example of archaic English sounding archaic, nothing in the Latin necessitates that translation. The phrase "gratias deo ago" would probably best be translated simply as "I thank God."

I don't think there are many clean cut examples of the dative of possession really bubbling up into English. I'm no linguistic historian by any means but I think one plausible place where it still echoes in English might be in phrasal verbs like "belong to" or "send for." I think any true use of the dative of possession in English is largely impossible because we rarely consider nouns to have any relationship with their verbs and instead rely on helpful prepositions and the like to describe the concepts included in many Latin cases.