Let $\mathbb{F}_q$ be a finite field. Do there exist smooth projective varieties over $\mathbb{F}_q$ of arbitrarily high dimension that have no $\mathbb{F}_q$-points and no non-constant maps to lower-dimensional varieties?
Two lines of attack that don't work: Brauer-Severi varieties (the Brauer group of $\mathbb{F}_q$ is trivial) and varieties that become simple abelian over $\mathbb{F}_q$ (they have a rational point to begin with as mentioned by Wojowu).
Sure.
By Poonen's Bertini theorem, for every $q$ and $n$, the limit as $d$ goes to $\infty$ of the proportion of degree $d$ hypersurfaces in $\mathbb P^n$ that are smooth and have no $\mathbb F_q$-points is positive.
Being smooth hypersurfaces in $\mathbb P^n$ for (let's say) $n>3$, they have Picard rank $1$, and thus no maps to lower-dimensional varieties.
For an explicit smooth hypersurface of dimension $n-2$, take $\alpha \in \mathbb F_{q^n}$ with $\operatorname{tr}(\alpha)\neq 0$. Choose a basis for $\mathbb F_{q^n}$, so that elements of $\mathbb F_{q^n}$ can be written as vectors over $\mathbb F_q$, with multiplication given by an $n$-tuple of quadratic polynomials.
Consider the function $$ \operatorname{tr} ( \alpha x^{ \frac{q^n-1}{q-1}})$$ from $\mathbb F_{q^n}$ to $\mathbb F_q$. In coordinates, this is a homogeneous polynomial of degree $\frac{q^n-1}{q-1}$ in $n$ variables, hence defines a projective hypersurface, and sends every nonzero element of $\mathbb F_{q^n}$ to a nonzero element of $\mathbb F_q$, hence has no $\mathbb F_q$-points.
Over $\overline{\mathbb F_q}$, we can choose our coordinate system to be the $n$ embeddings of $\mathbb F_{q^n}$ into $\overline{\mathbb F_q}$, and in this coordinate system the equation is $\sum_i c_i x_i^{ \frac{q^n-1}{q-1}}$ where $c_i \neq 0$ is the image of $\alpha_i$ under the $n$th embedding, so this hypersurface is also smooth.
Just take the Weil restriction with respect to $\mathbb{F}_{q^r} / \mathbb{F}_q$ of a pointless curve. This will typically admit no $\mathbb{F}_q$-morphism to a variety of lower dimension.
Not an answer, this is just a (cw) out-of-curiosity example of the recipe from Will Sawin's answer. The following polynomial defines a smooth projective surface without any $\mathbb F_3$-points: $$\scriptstyle -t^{40}-zt^{39}-x^3t^{37}-x^4t^{36}+y^4t^{36}-xy^3t^{36}-yz^3t^{36}-x^3yt^{36}-x^9t^{31}+y^9t^{31}-z^9t^{31}+x^{10}t^{30}-y^{10}t^{30}-xz^9t^{30}-yz^9t^{30}-x^9yt^{30}-y^{12}t^{28}-z^{12}t^{28}-x^3y^9t^{28}+x^3z^9t^{28}+x^9y^3t^{28}+x^9z^3t^{28}+x^{13}t^{27}-z^{13}t^{27}-xy^{12}t^{27}+yz^{12}t^{27}-x^3y^{10}t^{27}+x^4y^9t^{27}+y^4z^9t^{27}+xy^3z^9t^{27}-x^3yz^9t^{27}+x^9y^4t^{27}-x^9z^4t^{27}+y^9z^4t^{27}+x^{10}z^3t^{27}+y^{10}z^3t^{27}-xy^9z^3t^{27}+x^9yz^3t^{27}+x^{12}yt^{27}-x^{12}zt^{27}-y^{12}zt^{27}-x^9y^3zt^{27}-x^{27}t^{13}-y^{27}t^{13}-z^{27}t^{13}-xy^{27}t^{12}+xz^{27}t^{12}-yz^{27}t^{12}+y^{30}t^{10}+z^{30}t^{10}-x^3y^{27}t^{10}+y^3z^{27}t^{10}-y^{27}z^3t^{10}-x^{31}t^9-y^{31}t^9-z^{31}t^9-xy^{30}t^9+xz^{30}t^9+yz^{30}t^9+x^3y^{28}t^9-x^3z^{28}t^9-y^3z^{28}t^9+x^4z^{27}t^9+y^4z^{27}t^9+x^3yz^{27}t^9-x^{27}y^4t^9-x^{27}z^4t^9-y^{27}z^4t^9+xy^{27}z^3t^9-x^{30}yt^9-x^{30}zt^9+x^3y^{27}zt^9+x^{27}y^3zt^9-y^{36}t^4-x^9y^{27}t^4-x^9z^{27}t^4+y^9z^{27}t^4-y^{27}z^9t^4-x^{37}t^3+y^{37}t^3-z^{37}t^3-xy^{36}t^3+xz^{36}t^3+yz^{36}t^3-x^{10}z^{27}t^3-xy^9z^{27}t^3-x^{27}z^{10}t^3+y^{27}z^{10}t^3+x^{28}y^9t^3-x^{28}z^9t^3+y^{28}z^9t^3-x^{27}yz^9t^3+x^{36}yt^3-x^{36}zt^3-y^{36}zt^3-x^{27}y^9zt^3-x^{39}t-y^{39}t+z^{39}t-x^3y^{36}t+x^3z^{36}t-x^9z^{30}t-y^9z^{30}t-x^{12}z^{27}t+y^{12}z^{27}t-x^3y^9z^{27}t+x^9y^3z^{27}t-x^{27}y^{12}t-x^{27}z^{12}t+y^{30}z^9t-x^3y^{27}z^9t-x^{36}y^3t-x^{36}z^3t-y^{36}z^3t+x^{27}y^9z^3t+x^{40}-y^{40}+z^{40}-xy^{39}+yz^{39}-x^4z^{36}+xy^3z^{36}-x^9y^{31}-x^9z^{31}+y^9z^{31}+x^{10}z^{30}+xy^9z^{30}-x^{12}y^{28}+x^{12}z^{28}+y^{12}z^{28}-x^{13}y^{27}+y^{13}z^{27}-xy^{12}z^{27}-x^4y^9z^{27}-x^9y^4z^{27}-x^{10}y^3z^{27}+x^{12}yz^{27}-x^{27}y^{13}+x^{27}z^{13}+y^{27}z^{13}+xy^{27}z^{12}+x^{30}y^{10}-x^{30}z^{10}-y^{30}z^{10}-x^3y^{27}z^{10}+x^{27}y^3z^{10}-x^{31}y^9+x^{31}z^9+y^{31}z^9-xy^{30}z^9+x^3y^{28}z^9-x^4y^{27}z^9+x^{27}y^4z^9+x^{36}z^4-x^9y^{27}z^4+x^{27}y^9z^4-x^{37}y^3-x^{37}z^3+y^{37}z^3-xy^{36}z^3-x^{27}y^{10}z^3-x^{28}y^9z^3+x^{36}yz^3+x^{39}y+x^{39}z-y^{39}z+x^9y^{30}z-x^{12}y^{27}z-x^{30}y^9z+x^{36}y^3z $$ (this is the trace of $(x+y\alpha+z\alpha^2+t\alpha^3)^{40}\alpha$ where $\alpha$ is the generator of the multiplicative group of $\mathbb F_{81}$ and $x,y,z,t\in\mathbb F_3$).
In fact, using that $x=x^3=x^5=...$ and $x^2=x^4=x^6=...$ for any $x\in\mathbb F_3$, one can turn the above into a much shorter example: $$ -t^4-t^3 x+t^3 y+t^2 x z-t^2 y^2-t x^3-t x^2 y+t x^2 z+t x y^2+t y^3-t y z^2+t z^3+x^4+x^3 y-x^3 z+x^2 z^2+x y^2 z-x z^3-y^4-y^3 z+z^4 $$ However the latter might be non-smooth, as Will Sawin points out in the comment.
