I tested this further, and now I am utterly confused.
I had an idea, that the direct damage gets blocked by the shields, and only the afterburn takes effect on the unit behind, but this is certainly NOT the case.
Attacking single shieldbots (Convict and Thug) with Pyro:
The shieldbots began to take health damage at different shield levels, sometimes immediately, sometimes at about 60% to 50%.
The afterburn also took effect at different shield levels, but at 40% it was always present.
I tried multiple times both on Thugs and Convicts, and the results were inconsistent on both.
Recon and Guardian Commanders with flamethrower upgrade:
For the first time I could not reproduce the damage to the last unit (Mason), however it was with Commander flamethrowers with 8 range upgrades.
With the similar setting as in the image above, but adding 1 more shieldbot horizonrally (shields did not overlap), the Recon commander did no damage to the Mason.
Guardian Commander with dual flametrhowers never did damage to the unit behind the shieldbot in my test, only after the shield was depleted.
Adding multiple shieldbots:
Tested with Pyro, Thug, and Mason
I had one setup as the image above (except a Thug was in the Convict's place), and one similar, but with 3 Thugs vertically, close as possible to each other. And a Mason behind horizontally, like in the image. Paused the game and ordered the Pyros to attack the Masons, then unpaused. The Mason behind the 3 Thugs took more damage and died sooner than the one behind the single Thug.
I tested this only once.
All was done in singleplayer skirmish with cheats. Map: Daybreak Canyon 1.1
The previous test were on Titan Duel 2.2