This falls into a general class of hash functions known as homomorphic hash functions.
Your question is not entirely clear about what definition you are using for $+$. If you want the hash function to satisfy $h(x+y)=h(x)+h(y)$ where both instances of the $+$ operator refers to addition in a suitable group (e.g., addition of integers, xor), then such a hash is called linear. A CRC hash is an example of a linear hash function.
It's important to distinguish concatenation from addition. If you are looking for a hash that satisfies
$h(x||y) = h(x)+ h(y)$,
where $||$ denotes concatenation and $+$ denotes addition, then that's a different beast. I don't know of any standard name for it. This is a very restrictive property. Such hash function will always necessarily have the form
$h(x) = \sum_i h(x_i),$
where $x_i$ is the $i$th character of the string $x$. Consequently, the only way to achieve such a property is as follows: you choose some lookup table that maps from a single character to an integer; then you apply this to all the characters in the string and sum them up. Unfortunately, such a hash is less than ideal, as a hash function, as any two strings that have the same characters in a different will yield the same hash value. Thus if you use this as a hash function for a hash table, you might find an increased rate of collisions. However, it would be easy to construct such a hash function.
There are other generalizations. For instance, maybe you want a hash function with the property that
$h(x||y) = F(h(x), h(y))$
where $F$ is some associative function. There are standard constructions of such a hash. For instance, CRCs achieve this (though here $F$ depends on the length of $x$).
I suggest you also look at rolling hashes, the Rabin-karp rolling hash, Buzhash, AdHash, and MulHash. See also the following resources: