Burn out reduces the shields. If it does sufficient damage to burn through the shields, it will do damage with the remaining.
Example:
A Li Halan Iskati frigate fires at a Hawkwood frigate (hull 4, 3 shields)
The attack: lasers - 5AD, accurate, precise, burn out.
Roll 5 AD. 1,3,3,4,5: 4 hits.
The first three burn out the shield. The last one rolls on the attack table to inflict damage.
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Double burn reduces the shields by two for each hit.
Example: a Decados Tupot'ka destroyer fires at a Hazat frigate (shields 3).
It scores 3 hits with its EM pulse guns.
The first hit will burn out 2 shields. The second hit would knock out two, but there is actually only one shield left. The third hit inflicts damage.
Now the destroyer can open up with its slug guns on an unshielded ship!
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Example 3, just for completeness.
The Tupok'ta fire at a frigate and misses compltely with its EM pulse. It then fires its slug guns - 6AD accurate multihit 3.
It gets 3 succesful rolls - for a total of 9 hits. The first 3 hits are stopped by the shields (which are not reduced), 6 hits are rolled on the attack table. The target is hurt, but is still shielded.