What is known about ordinary character values at involutions?

There are many results about the values of $\chi(t)$ when $t$ is an involution of a finite group $G$ and $\chi$ is an irreducible character: Isaacs' book on Character Theory has many such results collected from the literature, but there are many others scattered around:

For example, if $G = O^{2}(G)$ (equivalently, if $G/G^{\prime}$ has odd order), then $\chi(1) \equiv \chi(t)$ (mod $4$).

(Knörr): We have $\chi(t) = 0$ for every involution $t$ if and only if $|S|$ divides $\chi(1)$, where $S$ is a Sylow $2$-subgroup of $G.$

Regarding block theory, whenever $t $ is an involution of $G$ and $\chi$ is an irreducible character in the principal $2$-block of $G$, we have $\chi(tuv) = \chi(tu)$ whenever $u,v \in C_{G}(t)$ have odd order and $v \in O_{2^{\prime}}(C_{G}(t)),$ which is a consequence of Brauer's Second and Third Main Theorems.

I could probably give several more examples if you gave further clues as to what you are looking for.

Further edit to address a question from comments: if $t$ is an involution of $G$ and $B$ is a $2$-block of $G$, then results of Brauer imply the following facts (among others):

If $B$ has defect group $D$ and $t$ is not $G$-conjugate to an element of $D$, then we have $\chi(t) = 0 $ for every complex irreducible character $\chi \in B$.

If $B$ has dfect group $D$ and some conjugate of $t$ lies in $D$, then there is an irreducible character $\chi \in B$ with $\chi(t) \neq 0$, and we have $\sum_{ \chi \in {\rm Irr}(B)} \chi(1)\chi(t) =0,$ so there are irreducible characters in $B$ taking both positive and negative values at $t$. Later edit: Another theorem of Brauer is that if $B$ is a $2$-block of defect $d >1$, then the number of irreducible characters in $B$ of degree exactly divisible by $2^{a-d}$ is divisible by $4$. In particular, this implies that if $|G|$ is divisible by $4$ and $t$ is an involution of $G$, then the number of irreducible characters $\chi$ in the principal $2$-block such that $\chi(t)$ is odd is a multiple of $4$.

For the symmetric group, let $\chi^\lambda$ be the symmetric group charater canonically labelled by the partition $\lambda$. Then $\chi^\lambda(x) = 0$ whenever $\lambda = \lambda'$ is a self-conjugate partition and the involution $x$ has an odd number of disjoint transpositions. This isn't very deep: in fact $\chi^\lambda(x) = 0$ for any odd permutation $x$. The converse does not hold: for example $\chi^{(6,3,2,2,2)}(1,2) = 0$.

As a very weak sufficient condition, it follows easily from the Murnaghan–Nakayama rule that if all the $2$-hooks in the partition $\lambda$ are horizontal (i.e. two boxes in the same row) then $\chi^\lambda(1,2) \ge 0$, with strict inequality unless $\lambda$ is a $2$-core (i.e. a staircase partition as in Saxl's Conjecture).

If the involution has no fixed-points, then the Murnaghan-Nakayama rule is cancellation-free. Hence, the character value is (up to a sign) the number of domino tableaux of the shape $\lambda$. The number of such tableaux can be computed via a hook-formula (Fomin-Lulov / James-Kerber).

You can extend this to all involutions, but you need to sum over all possible ways to distribute single boxes in $\lambda$, so that they occupy some skew shape $\lambda/\mu$, then apply the above argument for each shape $\mu$.