Can I cast Mizzium Mortars for its alternate cost with Fires of Invention?
The Gatherer page for Fires of Invention says (emphasis mine):

You may choose to pay the mana cost or an alternative cost for spells you cast, even if their converted mana cost is less than or equal to the number of lands you control.

as well as

If you cast a card “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs (such as that of Fling), you must pay those to cast the card.

So yes, you can cast an overloaded Mortars, but the Fires will not help you to pay for it.