(In collaboration with Z. Chase.)
A Fibonacci number $F_{n}$ is never sandwiched between two twin primes $(p,p+2)$.

This is because this would require $F_{n}+1$ to be a prime, but that can only happen iff $n=1,2,3$, and one can check that $F_{n}-1$ is not a prime in these cases.

The fact that $F_{n}+1$ is a prime iff $n=1,2,3$ is probably quite old. One reference is [this OEIS page][1], where (the) Richard Guy shows

 1. $F_{4n}+1 = F_{2n-1} L_{2n-1}$,
 2. $F_{4n}+1 = F_{2n+1} L_{2n}$,
 3. $F_{4n+2}+1 = F_{2n+2} L_{2n}$,
 4. $F_{4n+3}+1 = F_{2n+1} L_{2n+2}$

where $L_n$ is the $n$th Lucas number.


  [1]: https://oeis.org/A001611