Clearly, their sum is greater than $2$, thus their sum is odd. Since all three numbers cannot be odd (since their product is $80$), then two of the numbers are even and one is odd. There are only two odd divisors of $80 = 2^4 \cdot 5$: $1$ and $5$. Consider these cases.

$c=1$, the following is possible:

$a = 2$, $b = 40$, $a+b+c = 43$ is prime.
$a = 4$, $b = 20$, $a+b+c = 25$ is not prime.
$a = 8$, $b = 10$, $a+b+c = 19$ is prime and less than $43$.

$c=5$, the following is possible:
$a = 2$, $b = 8$, $a+b+c = 15$ is not prime.