5. B.

Take out a coin and place it elsewhere, then divide the remaining coins into two groups $A$ and $B$ (each group has 1011 coins), and let $W(N)$ represent the total weight of the coins in group $N$.

For the first weighing, compare $W(A)$ with $W(B)$. Assume $W(A) \geqslant W(B)$, then divide group $A$ into three subgroups $C$, $D$, and $E$ (each subgroup has 337 coins), and perform two more weighings to compare $W(C)$ with $W(D)$, and $W(D)$ with $W(E)$. The following discussion is divided into two cases.
(1) When $W(A)=W(B)$, group $A$ and group $B$ each contain one counterfeit coin. Among the three subgroups $C$, $D$, and $E$, there is one subgroup that has a different weight from the other two. This subgroup is heavier, so the counterfeit coin is heavier. Otherwise, the counterfeit coin is lighter.
(2) When $W(A)>W(B)$, then only one of the groups $A$ or $B$ contains a counterfeit coin.
(i) If $W(C)=W(D)=W(E)$, then group $A$ does not contain a counterfeit coin, and group $B$ contains one or two counterfeit coins. Therefore, the counterfeit coin is lighter.
(ii) If $W(C)$, $W(D)$, and $W(E)$ are not all equal, then group $A$ contains at least one counterfeit coin, and group $B$ does not contain a counterfeit coin. Therefore, the counterfeit coin is heavier.