Is heathland a climax community?
Or is it a sub-climax sere, that will proceed to woodland by succession?
If the latter is true, are all efforts to conserve heathland futile?

I would start by asking what you mean by a climax community- it's usually inferred when imposing a human timescale on a vegetation community process, which of course is usually irrelevant in the context of the vegetation.
Secondly I wonder what kind of heathland you are thinking about?  In the UK much heathland would, indeed become invaded by trees in the absence of grazing, but that might be over very many years in some contexts where soil conditions are at their most challenging for tree growth.  Elsewhere, including in boreal and tropical contexts, heathland may be a stable community in its own right.
sorry for the potentially confusing answer, but I think it makes it clear that simple statements about successional trajectories are often problematic!