Suppose I've awakened a creature, using Awaken. If it mated, and had offspring, would the offspring have the same intelligence as the awakened parent creature?
Or, alternatively, if the awakened creature was to take time to educate the offspring, could it result in the offspring becoming intelligent? After all, the original creature speaks the same language as its offspring.
I believe the answer is very likely to be "no." The following is my rationale.
There is nothing in the description of the spell which indicates a change in the race or species of the target. The target itself is awakened and its game features change. The offspring would be of the same race or species as the original target, which has not changed, and so the awakened state would not be passed down to the offspring.
There is no indication in the rules that a creature's six ability scores are inherited from its ancestors. The closest analogue is that creatures of a humanoid race inherit a small modifier to one or more ability scores. At most, we might be able to infer that an awakened plant's offspring would have a boost of +2 to Intelligence, which would not be enough for them to act as if awakened and which would require us to rule that the spell is capable of creating a new race, a concept that is discussed below.
The heritable nature of spell effects would be a very major feature to be only implied by a spell description and not explicitly stated. We would expect that the power to use a single casting of a 5th-level spell to spontaneously create a new race or species with modified game features and then propagate an entire clade of them would be world-changing enough to be given at least a cursory mention in the rules.
Finally, if we ruled "yes," then that would imply that many other spell effects can be passed down through heredity in the absence of any clear indication that they can or cannot, creating a slippery slope. In addition, we would be left wondering why there are not more naturally-occurring awakened plants when there are a grand total of two awakened creature stat blocks in the SRD, and those two explicitly mention that the creature has been awakened directly by means of magic.
I don't think it is terribly opinion-based, then, to conclude that in the absence of any explicit statement on spell effect heredity it is unlikely that it was an intended implication of the rules. Of course, the DM could follow the "rule of cool" and allow isolated occurrences if it makes the narrative more enjoyable, or indeed they could use it as a justification for the creation of an entire race of awakened creatures in a campaign setting of their design, but either way they would risk the slippery slope I mentioned above.
If you're talking about genetic inheritability, we run into a tricky situation. The spell's duration is "instantaneous," indicating that the spell's effect is permanent and not reliant upon continuing magical energy to remain in effect. In other words, the spell creates a permanent, non-magical change in the creature.
For that to happen to an animal, it would have to grow entirely new brain structures that somehow still fit inside its skull - we'll just assume that happens however it happens. However, now you're talking about an animal that has a radically different physiology than others of its species. If the physiological change is represented in its DNA (which would be necessary), and that change transmits to its gametes (unknown), then it may not be able to successfully mate anymore - or, if it could, the resulting brain structure in the offspring would be . . . troubling, to say the least.
Additionally, you'd also have to create enough awakened creatures to allow for the required genetic diversity the species needs to successfully propagate. That differs for each species, but you're probably looking at several dozen for most mammals. Also, you have to make sure they don't mate with ordinary members of their species, or else the trait would be watered down and likely lost over time since there would not be much selective pressure to maintain the trait.
So, long story short, if your DM rules that the DNA changes apply to the creature's gametes, then sure, so long as you awaken several dozen other members of its species and keep them isolated from ordinary, non-awakened members long enough to ensure their propagation - a process that could take decades, or centuries, or longer.
