Leontief input output model with column sum greater than 1
In terms of if and only if statements according to Peterson &amp; Olinick (1982);

A substochastic matrix A is productive if and only if $I-A$ is
nonsingular.

In substochastic matrix the sum of entries by row or columns will not be greater than 1 so it is part of the condition but in addition the matrix $I-C$ should also be nonsingular. The nonsingularity is important for the invertibility of the matrix.
Consequently, I do not think that there are exceptions where the industries are unprofitable but the matrix is still productive. Rather there are exceptions where just because the entries sum to less to 1 and matrix is non-negative but it is singular and so we cannot invert it, in which case it would not satisfy conditions to be productive.