The answer to that is yes. AI ships will eventually run out of fuel when using fuel-consuming weapons.
I actually first saw evidence this in EV Classic... I had used a plug-in to change the Bounty Hunter to a Shuttle. However, I gave that Shuttle a Particle Beam, which ate fuel like a glutton. The Bounty Hunter would fire it at me for a time, but eventually run out of fuel and do nothing more than charge at me harmlessly.
You could try that Thunderhead Lance experiment to prove it if you wanted. Set a particular variant so that it has only Thunderhead Lances, 1 jump of fuel, no fuel-regen, and no other weapons to annoy you with. Then, set it to always show up when a pirate düde is called, and a system to always call that düde. With such a plug-in, you could go to that system, and observe the Thunderheads to see if they eventually run out of fuel and stop firing.
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"Trust me, this'll work..." - Me.
"WARPCORE BREACH IN 10 SECONDS." - Computer.
"... Of course, I could be wrong..." - Me.