As we should all be aware by now, the Starcraft lore has changed substantially during development and over the course of the games and expanded universe materials.
For example, the '96 website archive shows that the border sector Koprulu was originally meant to be in contact with the rest of Terran civilization across their space and Arcturus was originally a much more idealistic figure.
But the focus of this topic is the inconsistencies in the original manual. If you carefully read the manual and cross-reference different mentions of the same concept, you will notice that different passages contradict one another as if they originate from different drafts.
For example, overlords are mentioned in the history of the zerg as being present on Zerus whereas the unit biography states their ancestors Gargantis proximae were a space-faring race. This introduces an inconsistency because the zerg assimilated the space-faring behemoths to escape Zerus when they already had assimilated the G. proximae. Reconciling this would require positing explanations like there being two different breeds that were called overlords or that the G. proximae that came to live on Zerus had lost their spaceflight capability.
There are a number of other examples as well.
Khaydarin crystals are mentioned as being used by the xel'naga as the key component of their experiments. It is mentioned that only organisms thus engineered will be compatible with the crystals' interface. However, other details contradict (my rationale is provided afterward):
1) The protoss history states that crystal-compatible life is rare and powerful, even though earlier in the same chapter the xel'naga are stated to have experimented on countless species across the galaxy.
Rationale: the number of species who can utilize the crystals in their technology and such are rare, not merely compatible. The protoss were surprised because the zerg probes were running crystal-compatible software (wetware?). It is like humans discovering that aliens are running Linux.
2) The biography of the pylon states that the "sacred" crystals are only found on the protoss home world, even though they were used across the galaxy by the xel'naga.
Rationale: The best crystals for pylons are manufactured on the protoss home world, but lower quality crystals are naturally occurring on other worlds.
3) The zerg incorporate the "energies" of the crystals after consuming the xel'naga, even though they should already be compatible.
Rationale: The zerg incorporated the physical crystals (as seen in the briefing screens) and the underlying operating systems (as seen by the automatic translations into protoss language) into their physiology. This was easy because they were already compatible with the radiation.
However, other contradictions are more difficult to reconcile.
The protoss chapters cannot agree whether the protoss are a galactic empire or limited to a single planet. Their history mentions that they colonized hundreds of worlds over centuries after instituting the Khala, and they made contact with advanced civilizations and watched over countless primitive civilizations. Other passages are written as though the protoss only live on Aiur, such as their unit biographies. There is no way to reconcile this (the games claimed that this empire collapsed, but that undermines the entire premise because the zerg were searching for a tool that would allow them to fight the otherwise unstoppable empire on their own terms, at least until that was retconned into incoherence).
The brief biography of the Kel-Morian Combine's pirate militia makes no mention of the Guild Wars explained in the history section. This is important because the Guild Wars resulted in the Combine losing their affiliated guilds to the Confederacy and being turned into a client state. In the biography they are stated to have close ties to the Confederate government and are given special privileges, but you would expect them instead to be kept at arm's length and plotting revenge (e.g. take advantage of the First Contact War).
Those were the inconsistencies that I found so far, but there may be others that I missed.
Feel free to discuss, critique, advise, etc.