The answer is no: for example, $k[[x]]\otimes_k k((x))$ is not noetherian.  
Indeed, if it were, so would be $ k((x))\otimes_k k((x))$.   
But this would contradict the following interesting general theorem of Vámos:    

Given an extension of fields $K/F$ the tensor product $K\otimes_F K$ is noetherian if and only if the $K$ is finitely generated as a field over $K$.  

**Full confession**  
 I have only  read an abstract of Vámos's  article because I have no access to it. Anyway, here is the reference:      
P. Vámos, On the minimal prime ideal of a tensor product of two fields, Math. Proc.
Cambridge Philos. Soc. 84 (1978), no. 1, p.25-35.