When does sum of two units give a unit? Let $A$ be a ring and $u,v \in A^\times$. When do we have that $u + v \in A^\times$?

I think that A is needed to be an integral domain. For example consider $\mathbb{Z/6}$. Both $1$ and $5$ is a unit but their sum $1+5=0$ is not a unit.
I do not think there is such a condition. For any unital ring such that $1 \neq 0,$ $1$ and $-1$ are units, but $1+(-1)=0,$ which is not a unit.
In any field, the sum of two units is a unit, unless they're additive inverses.  For a non-field example, let $\zeta$ be a primitive sixth-root of unity and let $A$ be the ring of integers in $\mathbb{Q}[\zeta]$. Then note that $\zeta^2+1 = \zeta$.  (Because $0 = \zeta^6 - 1 = (\zeta^3-1)(\zeta^3+1) = (\zeta-1)(\zeta^2+\zeta+1)(\zeta+1)(\zeta^2-\zeta+1)$ and we can set this last factor equal to $0$.)