
A cut stump at or above grade with the original root system intact below it. Over time: mushrooms or shelf fungi on the top and sides, bark separating from the wood, the surface going soft and punky, ants or beetles working the interior, and a sunken ring in the lawn as the roots break down. Plus a trip hazard and a mower you keep steering around.
Not urgent
A stump is a slowly decaying mass of wood, not a hazard. It is not going to fall and it poses no structural threat to a house. What it actually does is occupy the spot you probably want to replant, keep feeding decay fungi already present in the soil, and interfere with mowing. The one forward-looking concern is that some stump-colonizing root-decay fungi, Armillaria among them, can also attack living roots of nearby trees — so a stump acts as a reservoir rather than an inert object.
Normally a stump grinding job: $100 to $700 around Indianapolis, depending on size and access. A range, not a quote.