The answer is no: for example, $k[[x]]\otimes_k k((x))$ is not noetherian.
Indeed, if it were, so would be $k((x))\otimes_k k((x))$.
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 $K$. F$. Full confession I have only read an abstract of Vámos's article because I have no access to it. Anyway, here is the reference: P. Vámos, On the minimal prime ideal of a tensor product of two fields, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 1, p.25-35. 1 The answer is no: for example,$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\$.