So, let me start from my last game.  We closed with our party's Paladin finding (and accidentally activating) a cursed longsword that he cannot let go of.  Aside from the implications of that, the sword constantly talks...LOUDLY!!  Thus, the stealth has been basically ruined for the entire party (level 3).

Flash forward to this evening's game, and our Paladin player is unable to attend and I wound up running his character per his request...and he rolled very poorly, missing everything he tried to hit but remained unscathed.  After defeating the encounter, our Bard persuaded our Paladin to let him wrap and tie off the sword (and the paladins primary arm from the elbow down) into his Bag of Holding in an attempt to muzzle the sword, which is howling violently that it is afraid of the dark.  Fortunately, that was where our session ended.

QUESTION:  Would the effects of a Bag of Holding mute the sounds coming from the sword?  

I look forward to your responses, because I am at a loss as to how effective this would be...especially since there is a boss encounter in the very next room. 

Since the bag can't be completely sealed while his arm is sticking in it, some sound will probably still come out (basically passing through the arm).

It should still have a very strong dampening effect, at least not worse than heavy armor.

There's no rules for it so it's entirely a DM decision, but personally as a DM I would probably reward the creativity and allow it (or at least have it significantly muffle the sound).

For me, the simplest thing to do is to rule that his technique would give disadvantage to any stealth check made by the party rather than negating any chance of stealth.

Quote from Mog_Dracov >>

For me, the simplest thing to do is to rule that his technique would give disadvantage to any stealth check made by the party rather than negating any chance of stealth.

 Yeah, that would be my take as well. Alternatively, you could let them roll normally, and instead penalize the Paladin on his attacks somehow. Like he has to spend an action getting the sword free, which is basically RAW but feels bad, or maybe he has to make a Dex check to untie the rope fast enough, otherwise he uses an action doing that before he can spend an action drawing the sword.

Have they tried talking to the sword to rationalize with it or to find out a way to 'appease' it so it can cooperate?  This is assuming it is intelligent and not just shouting gibberish. DM should have a solution or allow reasonable approaches to not completely screw the party.

As long as you tighten the mouth of the bag around the arm then yeah, if they don't tighten it then I say just disadvantage for the paladin's check at most. It's a whole pocket dimension! It's just yelling into the ether (well, astral) and honestly not likely to be heard much anymore.