This seems like a matroid question but I'll take an elementary approach. It may be that my remarks just knock off the easy observations.
Before we even factor in the information gleaned from $M_I,$ there are restrictions on which maximal minors of $M_{I'}$ can vanish (and the same restriction apply to the information we can get about $M_I$). If a certain maximal minor is non-zero then there are these possibilities for each remaining row : Either it belongs to a non-vanishing minor with some $j-1$ of the previous rows -Or- it is the zero row and any minor it participates in vanishes. So your question is perhaps what else can we say given the additional information from $M_I?$ .
Extremes can be good to think about. Assume $I$ is the first $j$ columns. It will follow from below that
(*) if every minor for $M_I$ vanishes, the same is true for $M_{I'}$ and that
(* *) otherwise we may assume that the top minor is the $j \times j$ identity matrix and
(* * *) that we know which entries of $M_I$ are zero.
- Whatever is true for $M_{I}$ it could happen that every minor of $M_{I'}$ vanishes (if all the other columns are in the span of $I$).
- If $j=1$ then we know which entries in column $1$ are zero. A minor of $M_{I'}$ with all zero in the first column will vanish. Any other first column might or might not lead to a non-zero minor.
- In general, a minor for $M_{I'}$ with all leftmost $j \times j$ minors vanishing also vanishes (hence (*)).
Column operations on $M_{I'}$ will affect the values of the minors but not which ones vanish. And permuting the rows is just a relabelling which does not change the problem. Hence (* * ). And then just the minors which use $j-1$ of the top $j$ rows will tell you the rest of (* * *).
There is the question of what the vanishing pattern can be for $M_I$.
So your question is perhaps what else can we say given the additional information from $M_I?$
I suppose another extreme is $m=n+1$. We already saw that it might be that every minor of $M_{I'}$ vanishes and that otherwise we may assume that the top $j$ rows of $M_{I'}$ agree with the $(j+1) \times (j+1)$ identity matrix. We then know which entries of $M_{I'}$ are zero and which are not simply from (* * *) above and (in the final column) from the vanishing or non-vanishing of maximal $M_{I'}$ minors using the top $j$ rows.