(EDITED) You certainly can't have it for all $\theta$.  Consider $n=2$.  An eigenfunction
$f(\theta_1,\theta_2) = \exp(i k_1 \theta_1 + i k_2 \theta_2)$ for eigenvalue $\lambda = -k_1^2 - k_2^2$ will have $f(0,0) + f(\pi,\pi) = f(0,\pi) + f(\pi,0)$ unless $k_1$ and $k_2$ are both odd (i.e. $\lambda \equiv 2 \mod 4$), in which
case $f(0,0) + f(\pi,\pi) = -f(0,\pi) - f(\pi,0)$.