If I wanted to cast finger of death but I'm all out of 7th level spell slots, could I cast it using an 8th or 9th level spell slot, even though it doesn't change the spell's effect?
Yes, you can cast any spell with a higher level spell slot, regardless of whether it gains anything by being cast from the higher slot.
From the PHB chapter on Spellcasting, page 201:
When a character casts a spell, he or she expends a spell slot of that spell's level or higher, effectively "filling" a slot with the spell. You can think of a spell slot as a groove of a certain size - small for a 1st-level slot, larger for a spell of higher level. A 1st-level spell fits into a slot of any size, but a 9th-level spell fits only in a 9th-level slot.
So not only is this possible for any spell, the writers went to great lengths to describe the reasoning behind this.
Further down the page, we have:
Some spells, such as magic missile and cure wounds, have more powerful effects when cast at a higher level, as detailed in the spell's description.
This further calls out that only certain spells actually benefit from being cast from a higher level spell slot.