Depends on the DM, yet there is some established FR lore to guide us.
The answer by Quadratic Wizard is well-written for a generic D&D setting, but it was written before the question was tagged Forgotten Realms, so it does not fully address the specifics of the Forgotten Realms setting, where the fate of souls has been different from the generic treatments in the DMGs in most editions of D&D, including 5e.
According to the Sword Coast Adventurer's Guide (page 20):
Most humans believe the souls of the recently deceased are spirited away to the Fugue Plane, ... The servants of the gods come to collect such souls and, if they are worthy, they are taken to their awaited afterlife in the deity's domain. Occasionally, the faithful are sent back to be reborn into the world to finish work that was left undone.
Souls that are unclaimed by the servants of the gods are judged by Kelemvor, who decides the fate of each one. Some are charged with serving as guides for other lost souls, while others are transformed into squirming larvae and cast into the dust. The truly false and faithless are mortared into the Wall of the Faithless, the great barrier that bounds the City of the Dead, where their souls slowly dissolve and begin to become part of the stuff of the Wall itself.
These statements are in agreement with the lore established in the past editions of D&D (since at least the 2e, added by the TSR designers on to Greenwood's Realms, as you can read in his tweets here and here). For example Faiths and Avatars (2e) states:
A person's patron deity is the power that eventually escorts that person's spirit from the Fugue Plain, the place where spirits go after people die, to its afterlife as a petitioner in the Outer Planes in the realm (or at least the plane) of its patron deity.
In short, where the souls go, whether they were believers or not, is not an issue of alignment in the Forgotten Realms, it is about which deity collects them. The critical part that surely needs interpretation is whether unaligned creatures do have souls. Most unaligned creatures from the Monster Manual are animals (beasts) or monstrous creatures, while there are also elementals and constructs as well. One DM might argue that none of these have souls, while another might rule that animals have souls, while the the spirit of an elemental or the spirit that animates a construct is not really a soul and goes directly to the inner planes.
If you decide that the dead animal had a soul, it is likely be collected by the agents of Silvanus if it was a wild animal, or by the agents of Mielikki if it had been a companion of a ranger or other follower of Mielikki, or by the agents of Chauntea if it was domesticated. A particularly vicious animal's soul might also end up in Malar's domain, some deadly spiders might even go to Lolth's domain, as the DM sees fit.
It is worth noting that in 2e (the latest edition which had the 5e's Great Wheel Cosmology applied to the FR), the domain of Silvanus was on the Outlands, Mielikki's was on the Beastlands, Chauntea's was on Elysium, Malar's was on Carceri and Lolth's was in the Abyss. If no deity's agents pick up the souls, it is reasonable to assume that Kelemvor would send the souls with animal intelligence to the most relevant plane.