We refer to [1] for the notions used in this post.
The Grothendieck ring of a fusion category (over $\mathbb{C}$) is a fusion ring, but there are fusion rings which are not of this form (categorification problem), for example if there are only two simple objects $1,X$ (up to isomorphism) then the fusion ring is completely determined by a non-negative integer $n$ such that $X \otimes X \simeq 1 \oplus n X$, and it is a Grothendieck ring of a fusion category if and only if $n=0,1$; see [2].
Note that a fusion category is a finite semisimple tensor category. Now, if we drop out the semisimple assumption:
Question 1: Is there a fusion ring which is not the Grothendieck ring of a finite tensor category?
Question 2: Is there a fusion ring which is the Grothendieck ring of a finite tensor category, but not of a fusion category?
[1, Remark 4.9.2]: the Grothendieck ring of a finite tensor category is not always a fusion ring; for example if $X$ is the $2$-dimensional irreducible representation of $S_3$ over a field of characterisitic two, then $[X \otimes X^* : \boldsymbol{1}] >1$.
Bonus question: Is there a finite tensor category over $\mathbb{C}$ whose Grothendieck ring is not a fusion ring?
References
[1]: P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik; Tensor categories. Mathematical Surveys and Monographs (2015) 205.
[2]: Ostrik, Viktor. Fusion categories of rank 2. Math. Res. Lett. 10 (2003), no. 2-3, 177–183
Best Answer
Here are answers by Pavel Etingof (reproduced with his authorization):
Question 1: Is there a fusion ring which is not the Grothendieck ring of a finite tensor category?
Answer: Yes, for example the rings in Example 8.19 in
https://arxiv.org/pdf/math/0203060.pdf
given by $gX=Xg=X, X^2=X+\sum_{g\in G}g$ do not admit a categorification (even non-semisimple) when $|G|>2$.
Indeed, assume the contrary. Since $gX=Xg=X$, we get that $X^2$ both projects to $g$ and contains $g$ for any $g\in G$. Thus each g and hence $\oplus_{g\in G}g$ is a direct summand of $X^2$. So any category which categorifies this ring must be semisimple. But in Example 8.19 it is shown that this ring has no semisimple categorification.
Question 2: Is there a fusion ring which is the Grothendieck ring of a finite tensor category, but not of a fusion category?
Answer: I don’t know. In positive characteristic i think it is likely that such examples exist.
Bonus question: Is there a finite tensor category over $\mathbb{C}$ whose Grothendieck ring is not a fusion ring?
Answer: absolutely. For example see the book "Tensor categories",
http://www-math.mit.edu/~etingof/egnobookfinal.pdf
p.105 (small quantum $sl(2)$). In fact, this is the typical behavior. Moreover, if the category is pivotal factorizable and non-semisimple (as in the above example), this must happen by
https://arxiv.org/pdf/1703.00150.pdf
Namely, it has a simple projective object $X$, so $X\otimes X^*$ is projective and contains projective cover $P(1)$ as a summand. If the Grothendieck ring is a fusion ring then $[X\otimes X^*:1]=1$, so since the category is unimodular (as it is factorizable), we get $P(1)=1$, hence it is semisimple.