To compensate for the blue screen used during the filming of Tasha Yar's memorial, Beverley Crusher had to wear a greener uniform (instead of the normal blue). Likewise, Wesley Crusher's acting ensign uniform was slightly modified, replacing the blue shoulder stripe with a green stripe.
When Armus "removes" Geordi's visor forcefully to toy with him and Data, LeVar Burton's brown eyes are visible as the visor leaves his face (especially visible with freeze frame). The character of Geordi is blind, with a fully white eyeball instead of a colored iris/pupil.
When starting up the warp engine, Lt. Cmdr. Lynch sets the ratio for the matter-antimatter injectors at 25:1. Yet in Star Trek: The Next Generation: Coming of Age, it was established that there is only one ratio with matter-antimatter, 1:1.
When Worf is first describing the half-buried shuttle, the top portion of the monitor display (above the topographical map with the shuttle) shows the full energy graph, long before the discovery of the energy fluctuations is made. Then, after the fluctuations are discovered, the graph is shown at its beginning.
(at around 29 mins) When Picard crouches down while talking to Armus, the top of Armus' "head" can be seen rising out of the pool, in the lower right corner of the shot facing Picard. Then, in the immediate next shot facing the pool, Armus breaks the surface and rises again.
As Captain Picard walks up to the pool of slime for the first time, two white lights reflect off the surface. The same is also true in shots of Armus covering the shuttle. This has to be a reflection of studio lights, since there is no other sign of life on the planet and no other white light on the planet.
Geordi's phaser falls off into the oil slick-like creature (Armus) when he runs up to try and rescue Riker. However, it doesn't fall entirely into it but lands on the edge of the pool. Therefore, Geordi would have been able to pick it up again and clean it, which is why he later can be seen with his phaser back in its holster.
When Armus re-emerges from his pool for the first time, the slime can be seen bubbling right at the area where the mouth would be. (This is because the actor in the costume had to spit out the goo in order to breathe after coming to the surface.)
The goof items below may give away important plot points.
Incorrectly regarded as goofs
The cloud patterns behind Yar's farewell address alternate between close and far away, depending on whether the camera takes a close-up shot of Tasha or a distant shot. The different viewpoint of the camera naturally affects the appearance of clouds as well, whether they are holographic or real.