You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only commutes with limits in the second variable. Dually, you can think about homs as a kind of limit (in the second variable); you're asking the tensor product functor $(-) \otimes_A N$ to commute with this limit, but usually tensor products only commute with colimits.

This sort of reasoning not only suggests that your statement should be false but suggests what extra hypotheses might make it true: namely, some kind of projectivity hypothesis on $L$, or some kind of flatness hypothesis on $N$. In fact the statement is true if either $L$ or $N$ is f.g. projective; these conditions are equivalent to requiring that $\text{Hom}_A(L, -)$ commutes with all colimits or that $(-) \otimes_A N$ commutes with all limits respectively.

But it's also true if $L$ is finitely presented and $N$ is flat! In this case $\text{Hom}_A(L, M)$ is a finite limit (really an iterated finite limit, but this isn't an issue) in $M$, and $(-) \otimes_A N$ preserves it.

You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only commutes with limits in the second variable. Dually, you can think about homs as a kind of limit (in the second variable); you're asking the tensor product functor $(-) \otimes_A N$ to commute with this limit, but usually tensor products only commute with colimits.

This sort of reasoning not only suggests that your statement should be false but suggests what extra hypotheses might make it true: namely, some kind of projectivity hypothesis on $L$, or some kind of flatness hypothesis on $N$. In fact the statement is true if either $L$ or $N$ is finitely presented projective; these conditions are equivalent to requiring that $\text{Hom}_A(L, -)$ commutes with all colimits or that $(-) \otimes_A N$ commutes with all limits respectively.

But it's also true if $L$ is finitely presented and $N$ is flat! In this case $\text{Hom}_A(L, M)$ is a finite limit (really an iterated finite limit, but this isn't an issue) in $M$, and $(-) \otimes_A N$ preserves it. Dually, it's also true if $N$ is finitely presented and $L$ is projective: in this case $M \otimes_A N$ is a finite colimit in $M$, and $\text{Hom}_A(L, -)$ preserves it. Note that in Neil Strickland's example neither $L$ nor $N$ is finitely presented.