Must a CFII be instrument current to provide instrument training?
The requirements for providing instrument instruction are in 14 CFR 61.195(c) and they don't mention currency at all. There are also various FAA legal interpretations on whether or not CFIs (not CFIIs) require an instrument rating to provide instrument training (see this question), and what instrument training requires a CFII in the first place (see this question) and none of them mention currency either.

So no, there is no requirement for a CFII to be instrument current when providing instrument training.

Having said that, 61.65(d)(2)(ii) requires instrument students to do a cross-country under IFR with an instructor. An instructor who isn't instrument current couldn't provide that specific training flight, although obviously the student could use a different instructor. And many CFIIs like to take their instrument students into IMC to let them experience how different it is from being under the hood.

All in all, it's in a CFII's best interests to remain instrument current but it isn't required.