Why can't Zerg use Terran technology? They're intelligent enough to use technology. They have hive mind, Kerrigan, Terran-Zerg hybrids, etc.
Although it is never shown that Zerg use Terran technology, nothing speaks against them being able to do it. At least their "leader" types are very intelligent and thus should be able to use Terran technology (at least if there are no genetic locks).
So, even though they should be able to use it, they wouldn't do it anyway as they don't need to use Terran technology. It wouldn't give them any advantage in doing so. They have their own ammunition, which they can reproduce themselves (acid, spines, etc.) while Terran ammunition is limited. They can fly in space by themselves. So, they have no need to use it.
Furthermore I'm talking here about PURE Zerg, not infested. With infested...they are able to use Terran technology and do so.
So all in all: The Zerg CAN use Terran technology (at least via infested Terrans) and even do so. But Pure Zerg won't use it because they have no need for it (they do quite well without it).
Zerg are capable of using Terran technology, and it was shown to happen on multiple occasions:
This is an Apocalisk, which is an Ultralisk with integrated Terran Thor armour, courtesy of Alexei Stukov, an Infested Terran
Use their armour and their guns
Not sure how much technology is going on in there, but they work like normal bunkers, except for Zerg units.
Infested anti-air turrets
Capable of turning and firing their Terran weapons.
Infested Siege tanks
Capable of driving around, and turning the Siege Mode on and off.
Yes, these are all infested, by who is in control? The Zerg. Those are not Terrans inside anymore, they are Zerg now. A part of the Swarm. Zerg can assume direct control with Infestors (as opposed to indirect infestation), too, which is a gameplay mechanic.
Why don't the Zerg use Terran technology all the time? Hubris. The Swarm believes to be the apex of evolution, the most powerful faction. Why would they use inferior tools? As for Feral Zerg, well, they're Feral.
I would like to supply a more nuanced answer.
First of all, clearly, the Zerg do use terran technology. The infested uses terran technology.
Now, onto the question of "Why don't pure Zerg use terran technology?".
The most important reason is the anthropic principle and that our question kind of invites a tautology. The definition of "pure" zerg in this case would be a zerg that doesn't use terran technology. The definition of "infested" would be "zergs" that uses terran technology.
Infested are not just "infested humans". The image of this command center clearly shows that tenticle like things are growing out of it. The tenticle creature is a zerg, not based of humans, that uses terran technology.
Contact and Assimilation
Let's take a step back in time. Zergs had their first contact with Terran in Starcraft 1. As such, they only had about 4-5 years to adapt Terran technology to their own use. An alien specie, which never dealt with Terran technology before (or most other kind of non-biological), would be hard pressed to just adapt terran technology for themselves. It was mainly the infested terrans that made the assimilation of terran technology possible. As infested terrans, they would mostly be adapting terran technologies for themselves, and not for zerglings.
Terran Technologies are made for Terran, not zerg
How do you propose a zergling hold a machine gun? Why would mutalisks need a oxygen tank when it doesn't need to breath in space?
We adapt technology for ourselves; the zerg can adapt themselves for technology
Zergs can try to reverse engineer a machine gun, and adapt it so a zergling can use it. Or... they can create a new subspecies which is specially designed to use a machine gun.
In summary, a zergling doesn't drive a tank because a specially designed mass of tenticles with special bio-tech interfaces can drive a tank better. You just don't think of the tenticles as a pure zerg creature, because it uses human technology.
Other than being directed by 'Infested' Terrans like Stukov or Kerrigan, the Zerg Swarm doesn't interact (nor really care to interact) with Terran systems, preferring to stick to biological rather than technological evolution.
It has actually been considered (and likely tested) by Abathur, more than likely while the Queen of Blades (Infested Kerrigan) was around. Abathur and Primal Kerrigan discuss interfacing with Terran Technology in Heart of the Swarm, before the mission 'Friends Like These...':
Kerrigan: Is it possible for the Swarm to hack into the Dominion network?
Abathur: Terran systems, inelegant. Problem with interchange. Many factors obstruct request. Biology, language.
Kerrigan: So our thoughts are too different for anything beyond simple transmissions.
Abathur: Accurate. Zerg processes complex. Incompatible with terran machinery.
Kerrigan: Then it's up to Valerian now...
Starcraft Wikia: Starcraft II: Heart of The Swarm, 'Friends Like These...'
Emphasis mine. It stands to reason that other than repurposing Armor plating as covered in Andrey's answer, or simply using 'infested' structures as a place to produce infested terrans, interaction with Terran technology is out of the question.
They can and they do use Terran and Protoss technology. The Infestor's Neural Parasite is able to take control of other units and use their abilities.