What is the grammatical interpretation of the phrase? I don't understand what verb tense or voice is used.
I believe it is a subjunctive masquerading as an imperative. From the Wikipedia article on the subjunctive:
A present subjunctive verb form is sometimes found in a main clause, with the force of a third-person imperative (and such forms can alternatively be analyzed as imperatives). This is most common nowadays in established phrases, such as (God) bless you, God save the Queen, heaven forbid, peace be with you, truth be told, so be it, suffice it to say, long live..., woe betide... It can be found used more broadly in some archaic English.
Note that the "peace be with you" construction is most closely parallel to your "thanks be to God" example.
The authors of the ‘The Cambridge Grammar of the English Language’ recognize
. . . minor [clause] patterns, mostly involving fixed formulae or fragmentary structures.
Some of these clauses feature a subjunctive form of the verb, and Thanks be to God is one such example. Further examples are come what may and be it noted. David Crystal in ‘Rediscover Grammar’ calls this form ‘the formulaic subjunctive'.