Looking at your "variational" $\delta$-notation, it appears that it will be enough for you to consider the Gateaux derivative, which is a weaker version of the Fréchet derivative. For $F(p):=\int_X p\ln p\,dx$, the Gateaux derivative of $F$ at a point $p$ in the direction $h\in L^1(X)$ is $f'(0)$, where
$$f(t):=F(p+th).$$
For variational purposes, it would be even better to consider just the (right) directional derivative of $F$ at point $p$ in the direction $h\in L^1(X)$:
$$(D_+F)(p)(h):=f'_+(0):=\lim_{t\downarrow0}\frac{F(p+th)-F(p)}t$$
whenever this limit exists.
Even this right directional derivative will not in general exist for all $h\in L^1(X)$; e.g., it will not exist if the set on which $h<0$ and $p=0$ is of nonzero measure, because on that set $(p+th)\ln(p+th)$ will not be even be defined for any real $t>0$. However, noting that $(u\ln u)'=1+\ln u$ for real $u>0$, we see that, by the dominated convergence theorem, $(D_+F)(p)(h)$ will exist and be given by
the formula
$$(D_+F)(p)(h)=\int_X (1+\ln p)h\,dx$$
for any $h\in L^1(X)$ such that $|h\ln(p+th)|\le g$ for some $g\in L^1(X)$ and all small enough $t>0$.
