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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 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 $4$.

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 $4$.

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 $t$.

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 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 $4$.

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.

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 $t$.