Doesa cake filled with egg-less cookie dough filling have to be refrigerated?

Depends on what else is in it. What is the recipe?

I don't see anything that would require refrigeration. I just don't know if the dough chunks will turn to mush.

Chocolate Chip Cookie Dough Icing/Filling
Ingredients
FROSTING BASE:
1-3/4 cups confectioners sugar
1/2 cup unsalted butter, chilled
1/8 teaspoon salt
1/2 teaspoon vanilla extract
1 tablespoon milk
Eggless cookie dough (recipe below)

EGGLESS COOKIE DOUGH:
1/4 cup butter, softened
1/4 cup brown sugar, plus 2 tablespoons
2 teaspoons water
1/2 teaspoon vanilla
1/2 cup flour
1/4 teaspoon salt
1/3 cup mini chocolate chips (optional)

It should be fine. Doesn't need refrigeration for safety reasons. If it seems soft, add a little more flour to stiffen it up before you break it into chunks.

Even if it does get mushy in the heat, I wouldn't worry about it. I have a recipe that is basically everything you have listed there all whipped together. So it's kind of like a hybrid between buttercream and cookie dough, no chunks at all. No one's complained yet.