Which has the greater N–O bond length, NO₂⁻ or NO₃⁻?
There are two ways to think of it.  First: draw out the (major) contributing Lewis structures for both ions.  Each iin has one pi bond, but it's shared between two linkages in $\ce{NO2^-}$ ion versus three linkages in the $\ce{NO3^-}$ ion.  So the $\ce{NO3^-}$ ion has less pi bonding in each linkage making those bonds weaker overall.

Second: In all of those Lewis structuresthere are nonbinding pairs of electrons on the oxygen atoms, plus a pair on the nitrogen in $\ce{NO2^-}$.  There are more such pairs in $\ce{NO3^-}$ than in $\ce{NO2^-}$.  These pairs tend to repel each other and the bonding pairs, and $\ce{NO3^-}$ requires more room, thus longer bonds, to accommodate the greater number of closely packed nonbonding electron pairs.

These two factors tend to occur together when the molecules have nonnetalluc atoms that fill their votes, as here.