Do sodium channels keep renewing themselves with new subunits made by genes?
You are asking about two different processes: turnover (degradation) of mRNA and turnover of proteins.  In a typical mammalian cell the mRNA half-lives of stable messages approach the length of the cell cycle (e.g. 24 hr). One mRNA could be translated around the clock if needed. The resulting proteins will also have a measurable half-life, and that will vary with the protein and the cellular conditions. If a cell ran out of protein and its mRNA there is often a feedback loop that activates transcription of the structural gene back in the nucleus.