Roleplaying can get really fun when there's no escape from issues of personal hygiene and modesty.Apparently, also - at least per the OP - being inside a LTH makes personal ablutions and waste elimination a salient part of D&D play when it never comes up in any other circumstance of play that I'm familiar with.
Wizard: "And behold, Leomund's Tiny Hut!"
Rogue: "It's a dome."
Wizard: "Yes it is. A safe dome for us to sleep in tonight. Now relax and rest."
Rogue: "Where's the toilet?"
Wizard: "Toilet? Um, I guess it doesn't have one. Come on, let's get some sleep."
Rogue: "So what if we need to take care of you-know-what?"
Wizard: "Uh, I guess do it on the ground? Come on, it's not safe out there."
Rogue: "Screw this. I'm not sleeping next to somebody's steaming pile."
Wizard: "It'll be fine. The air is cleansed by the magic."
Rogue: "All fine until I turn over and roll onto a mess left behind by our barbarian here."
Barbarian: "Ooh, look, my hand passes through but the rock I pick up on other side doesn't come in."
Wizard: "We'll just make sure that all the piles are on the other side, okay?"
Rogue: "I'm still not taking my pants off in front of the paladin."
Paladin: "Hey, what's that supposed to mean?"