After davening Shachris I like to learn Mikra or Mishna with my Tefillin. But if I keep my Tallis on, I get hot and the distraction harms my learning. And if I sweat, it will stain the Tallis. (However, I know a wonderful Rav whose Tallis has yellowed from sweat over years of diligent and loving use, so maybe it’s a meritorious practice to allow the yellowing?)
Is it permitted or appropriate to take off and put away his Tallis while keeping Tefillin on?
When taking off our tallis and teffilin we sually take the tallis opff first and then the tefillin. However in your situation if you want to keep you tefillin on and take off your tallis you may. There is no need for you to purposely make your tallis get yellow and old looking, if anything it nmight be better to have a new nice looking tallis, because this makes the mitzva nice looking.