Is the sum of two prime ideals a prime ideal?

Th answer is no, consider the ring $\mathbb{Z}$ and the prime ideals $(2)$ and $(5)$. Since two and five are relative primes then every number can be written as a linear combination over $\mathbb{Z}$ of 2 and 5. To see this. note that if if $k$ is any interger, since we know that $$5-(2)2=1$$ by multiplying by $k$ we obtain the expression $$k5-(2k)2=k$$ which is a $\mathbb{Z}$-linear combination of $k$ in therms of $2$ and $5$. Note that $k5\in(5)$ and $-2k(2)=-4k\in(2)$. This means that $(2)+(5)=\mathbb{Z}$, which is not a prime ideal.

Note that the sum of ideals is an ideal when one is contained in the other. This is not possible in $\mathbb{Z}$ with the exemption when of the ideals is the prime ideal $(0)$, but other rings can give more interesting examples. Have a look at polynomials rings for example.