The primary substance of this question is answered in the Star Wars wiki article about blasters: (quotes edited for brevity)
...the blaster fired a compressed, focused, high-energy particle-beam
that was very destructive, commonly referred to as a "bolt."
Generating the bolt relied on two components: a gas cartridge filled
with an energy-rich blaster gas (typically Tibanna) and a power pack.
When the blaster was fired, a small amount of gas moved from the
cartridge... into the gas conversion enabler chamber... In this
chamber, the power pack energized the gas, before... the now extremely
high-energy gas was transformed into a compressed beam of intense
energy particles, coupled with intense light. The particle beam was
then focused... which generated a deadly high-energy particle beam,
fired from the emitter nozzle as a bolt of glowing energy.
The article on the DC-15A specifies that the particle beam takes the form of plasma:
Its power-charge magazines ionized the gas into charged plasma within
its ignition chamber. These bolts would then be accelerated out of the
rifle electromagnetically. The octagonal-like barrel would compress
the plasma into a thin bolt.
And the significance of Tibanna is explained in the article about Tibanna gas:
Tibanna gas produced four times its normal energy output when cohesive
light passed through it. Thus, when spin-sealed (compacted at the
atomic level), tibanna was used as a conducting agent in blasters and
other energy weapons, producing greater energy yields and thus greater
amounts of damage.
So Tibanna gas is not essential to fire a blaster, but it's greatly beneficial. And regardless of power output, blasters require SOME type of gas - it's what makes up the particles in the bolt. Without some gas remaining in the cartridge, there wouldn't be anything in the chamber to energize and focus into a high-energy bolt. Without any energy remaining in the charge pack, the blaster wouldn't be able to energize the gas particles enough to form a bolt.
More detail from the Tibanna article:
Most personal weapons could not tolerate this power boost, but
ship-mounted blasters benefited greatly from the use of tibanna gas.
Exceptions to this rule were the DC series rifles of BlasTech
Industries, which used tibanna gas to produce powerful ionized bolts
that were damaging to both organics and droids - something that was
crucial to winning the Clone Wars.
So, it may be reasonable to assume that the DC-15A rifle would function properly without any remaining Tibanna gas - but you'd have to replace it with a more mundane gas. Since other weapons of its type, outside of the DC series, use different types of gas, it may be possible to reload a DC rifle with non-Tibanna gas, but a gas cartridge designed for a different rifle might not fit. If you could reload a DC series rifle with another gas, the bolts it fires would be much less powerful. So if you run out of gas in your DC series rifle, you might as well toss it aside and switch to a smaller, lighter rifle that fires the lower-power bolts without Tibanna enhancement.
You also may be able to fire your blaster without any gas, but only set to "stun": (from the blaster article)
Most blasters also had a stun setting, which fed power through a
secondary emitter, bypassing the gas chamber to create a ring-shaped
electromagnetic burst that disrupted the target's nervous system,
often leaving them unconscious.
This wouldn't use any gas normally, so it wouldn't be affected by an empty gas cartridge.
In short: Gas is the ammunition which makes up the particle bolt, and the charge pack is what puts the bolt into a high-energy state. Tibanna gas is an enhancing agent - when it's used as the particle ammunition, it increases the rifle's power. Any blaster can theoretically work without it, but it would need to be replaced with a different gas, and if it's a high-power weapon you won't necessarily get the results you want when you fire it.