The integers $N$ with this property are those of the form $3m^2$ for some positive integer $m$.

In one direction, for $N = 3m^2$, the points
$$(m,m,m), (m,-m,-m), (-m,m,-m), (-m,-m,m)$$
form the vertices of a regular tetrahedron inscribed in the sphere $x^2 + y^2 + z^2 = N$.

Conversely, suppose that $P_i = (x_i, y_i, z_i)$ for $i=1,\dots,4$ are the vertices of an inscribed regular
tetrahedron. Then the center of this tetrahedron must equal the center of the sphere, namely $(0,0,0)$. Consequently, these four vertices together with $Q_i = (-x_i, -y_i, -z_i)$ for $i=1,\dots,4$ form the vertices of an inscribed cube in the sphere.
The side length of this cube is $(N/3)^{1/2}$, so its volume is $(N/3)^{3/2}$;
on the other hand, this volume also equals the determinant of the matrix
with row vectors $Q_2-Q_1, Q_3-Q_1, Q_4-Q_1$, which is an integer. Hence $(N/3)^3$ is a perfect square, as then is $N/3$.