How can I store leftover puff pastry or make it more durable? I am going to be attempting a recipe for tarts which use shop-bought puff pastry. I am fairly new to baking and this will be the first time I am using puff pastry. I am unsure what I can do with the remainder of the puff pastry once I have cut the circles that I need.

Can I just squash the piece together and stick it in the fridge for another time? Should I just bake them along with the tarts to use them up? Will they need to be thrown away?

My concern is the butter in the pastry and the fact that it will have warmed up by my rolling and handling and what impact that will have on subsequent use.
No, you probably don't want to just squish leftover puff pastry dough together to save it.  While it would be perfectly safe, it would ruin the layering that gives the rise and flakiness that is the reason for using puff pastry.

If you have a large enough section still intact, you could put in the refrigerator for a day or two, to use later.

Otherwise, a typical thing to do would be to prepare sweet or savory snacks with the surplus pieces.   Cut it into convenient shapes.  If you have large enough pieces, sticks would be good, even twisting them to give them an interesting texture.

Then, brush the pieces with butter (or oil), and sprinkle with cinnamon and sugar; or grated Parmesan cheese; or rosemary and salt; or some other complimentary flavor.

Bake the pieces up until they are puffed and golden brown and they are very nice snacks or treats.