As others have said......YES. Most certainly. Any portion of a wooden stock that is left unsealed will allow moisture level change in the wood - and that is what leads to shrinking & swelling, with the attendant accuracy problems. NEVER leave any portion of a wood stock unsealed, even the barrel channel.
Properly sealed, inside and out, even a solid wood stock is (relatively) impervious to weather changes. Unlike what some believe, it is NOT temperature change that causes problems with such stocks. The cause is change in the moisture content of the wood. That can be stopped with a thorough sealing job.
Minwax wood hardener may not be an adequate sealer. Can't say for sure, as I've never used this product. There are many things that will work, though. Tung oil (as you mentioned), sanding sealer, various varnishes, shellac, even paint. If you are finishing the outside with a Tung oil mixture, why not use it on the interior, too ? Make sure to flood all interior surfaces with sealer, even the trigger well.
In my experience, shellac is about the easiest sealer to use on the interior surfaces of a rifle stock. It is easy to apply (either spray or liquid) and dries very quickly.
