If the autocorrelation function of a zero-mean (stationary) Gaussian white noise process is $R_X(\tau) = E[X(t)X(t+\tau)] = K\delta(\tau)$, then the result $Y(t)$ of passing $X(t)$ through a filter with impulse response $h(t)$ is also a zero-mean stationary Gaussian process with autocorrelation function (and autocovariance function) given by $R_Y(\tau) = E[Y(\lambda)Y(\lambda+\tau)] = K\int_{-\infty}^{\infty} h(t)h(t+\tau)\, \mathrm dt.$
The above applies to Gaussian white noise as engineers understand the term. But, as I noted in this unanswered question, mathematicians also use the term (stationary) white noise to mean a process whose autocovariance function is given by $\text{cov}(X(t), X(t+\tau)) = \begin{cases} \sigma^2, &\tau = 0,\\ 0, &\tau \neq 0, \end{cases}$ in which case the description above does not apply. I would suppose that nonstationary white noise would mean $\text{var}(X(t))= \text{cov}(X(t), X(t))$ varies with $t$ instead of having fixed value $\sigma^2$ but I am not too sure about this.