Is $\lim_{t \rightarrow \infty} B_t = 0$ at equilibrium in classical monetary model, where $B_t$ represents bond holding? At equilibrium of classical monetary model, does $\lim_{t \rightarrow \infty} B_t = 0$, where $B_t$ represents bond holding has to hold? I was wondering about this question because there is no ponzi solvency condition in classical monetary model, and usually when inequality exists as part of constraints in economics model, they often can be represented as equality at equilibrium.
In general, no.

As you correctly say, No-Ponzi and Transversality imply that the NPV of real bond holdings go to zero. Since we (in general) have not pinned down inflation paths, this does not imply $\lim_t B_t = 0$, as you correctly suspect.

If you can provide more information about what model exactly you work on or the situation you're looking at, I could update my answer.