Bott vanishing holds for a smooth projective variety $X$ if $\mathrm{H}^j(X,\Omega_X^i\otimes\mathcal{L})=0$ for all $j\geq 1$, $i\geq 0$ and $\mathcal{L}\in\operatorname{Pic}(X)$ ample.
The Fano 3folds for which Bott vanishing holds (resp. fails) were classified by Totaro in Bott vanishing for Fano 3folds. Earlier he showed that del Pezzo surfaces of degree at least 5 satisfy Bott vanishing.
Fano threefolds satisfying Bott vanishing
ID  index  toric  description 

117  4  true 
projective space $\mathbb{P}^3$ 
226  1  false 
blowup of 115 in a line 
230  1  false 
blowup of 117 in a conic 
233  1  true 
blowup of 117 in a line 
234  1  true 
$\mathbb{P}^1\times\mathbb{P}^2$ 
235  2  true 
$\mathrm{Bl}_p\mathbb{P}^3$

236  1  true 
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ 
315  1  false 
blowup of 116 in the disjoint union of a line and a conic 
316  1  false 
blowup of 235 in the proper transform of a twisted cubic containing the center of the blowup 
318  1  false 
blowup of 117 in the disjoint union of a line and a conic 
319  1  false 
blowup of 116 in two noncollinear points 
320  1  false 
blowup of 116 in the disjoint union of two lines 
321  1  false 
blowup of 234 in a curve of degree $(2,1)$ 
322  1  false 
blowup of 234 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$ 
323  1  false 
blowup of 235 in the proper transform of a conic containing the center of the blowup

324  1  false 
the fiber product of 232 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$ 
325  1  true 
blowup of 117 in the disjoint union of two lines

326  1  true 
blowup of 117 in the disjoint union of a point and a line

327  2  true 
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ 
328  1  true 
$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ 
329  1  true 
blowup of 235 in a line on the exceptional divisor 
330  1  true 
blowup of 235 in the proper transform of a line containing the center of the blowup

331  1  true 
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

43  1  false 
blowup of 327 in a curve of degree $(1,1,2)$ 
44  1  false 
blowup of 319 in the proper transform of a conic through the points 
45  1  false 
blowup of 234 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ 
46  1  false 
blowup of 117 in the disjoint union of 3 lines

47  1  false 
blowup of 232 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ 
48  1  false 
blowup of 327 in a curve of degree $(0,1,1)$ 
49  1  true 
blowup of 325 in an exceptional curve of the blowup 
410  1  true 
$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ 
411  1  true 
blowup of 328 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(1)$curve 
412  1  true 
blowup of 233 in the disjoint union of two exceptional lines of the blowup 
51  1  false 
blowup of 229 in the disjoint union of three exceptional lines of the blowup 
52  1  true 
blowup of 325 in the disjoint union of two exceptional lines on the same irreducible component 
53  1  true 
$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ 
61  1  false 
$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ 