Direct sum seen as functor. We know that if $(M_i)_{i\in I}, (N_i)_{i\in\ I}$ and $(P_i)_{i\in\ I}$ are three families of $R$-modules, where $R$ is a ring with unity, then $$M_i\xrightarrow{f_i}N_i\xrightarrow{g_i}P_i$$ is an exact sequence if, and only if, $$\bigoplus_{i\in\ I} M_i\xrightarrow{\oplus f_i}\bigoplus_{i\in I}N_i\xrightarrow{\oplus g_i}\bigoplus_{i\in\ I}P_i$$ is exact.

I would like to know if it is true for an arbitray abelian category when the set $I$ is finite.
Yes, because homology is preserved by finite direct sums. If $M_i\to N_i\to P_i$
is a complex but not exact, it has nonzero homology $H_i$ at the middle terms. The direct sum $\bigoplus M_i\to\bigoplus N_i\to \bigoplus P_i$
will have homology $\bigoplus H_i$ at the middle term which doesn't vanish.

Of course if any of the $M_i\to N_i\to P_i$ isn't a complex then neither
is the direct sum.
Yes, because homology is preserved by finite direct sums. If $M_i\to N_i\to P_i$
is a complex but not exact, it has nonzero homology $H_i$ at the middle terms. The direct sum $\bigoplus M_i\to\bigoplus N_i\to \bigoplus P_i$
will have homology $\bigoplus H_i$ at the middle term which doesn't vanish.

Of course if any of the $M_i\to N_i\to P_i$ isn't a complex then neither
is the direct sum.