Should I refrigerate an opened box (TetraPak) of pumpkin puree?

it's not going to have a long life in your frig after it is opened so keep in the freezer(easy to scoop or dig out after softening a few minutes). If you like the flavor combo of peanut butter and pumpkin, consider that that combo can be found in a number of African dishes. Consider making a batch of soup w/: ~ sauteed onion and garlic, pumpkin and some pnut butter and a little tomato paste and chicken stock, or google for recipes.

you haven't said why you want it at room temp or when you plan to use it or for what recipe! if you plan to use it w/in 24 hours of opening it, it might be o.k. at room tempo, but i think it begins fermenting pretty quickly if not chilled. if you can, i'd advise chilling it and then removing it from the frig 2 hours before using it, if you have to work with it at room temp. And I am normally not very rule-following, btw.