After the pointy-shaped object leaves, the remaining blunt-nosed rocket will experience dramatically enhanced structural loading and possibly aerodynamic instability. They expect a "rapid scheduled disassembly" in flight, but if that does not happen they will either let it blow up when it hits the ocean, or blow it up as @RussellBorogove suggests if it violates termination criteria.
From CNET's NASA, SpaceX prepare for Falcon 9 rocket to explode over the Atlantic
It's likely to undergo what SpaceX fans refer to tongue-in-cheek as a "rapid scheduled disassembly." In other words, it's going to break apart and/or explode over the Atlantic Ocean.
According to the final environmental assessment for the test filed with the Federal Aviation Administration, a Falcon 9 carrying the Crew Dragon will launch from Florida's Kennedy Space Center and fly for about 88 seconds before the test is initiated. Once the Dragon separates from the Falcon 9 first and second stages, the rocket is expected to become uncontrollable and break apart.
From the linked environmental assessment, and as @AI0867 points out:
The launch scenario where an abort is initiated during the ascent trajectory at the
maximum dynamic pressure (known as max Q) is a design driver for the launch abort system.
Max-Q would also be the worst time for a powered rocket to loose its aerodynamic pointy-shaped nose.
The assessment continues:
2.1.9 BOOSTER DEBRIS DISPERSION
SpaceX anticipates a Falcon 9 breakup after Dragon abort. After thrust termination and abort separation, the Falcon 9 trajectory would be uncontrolled and would be expected to start departing from the nominal trajectory. In this nominal scenario, the propellant is expected to be consumed in the deflagration or aerosolized. This is consistent with behavior SpaceX observed in previous failures, including a Falcon 9 failure at their test site in McGregor, Texas, which failed at low altitude. In the event of an unanticipated and off-nominal condition, the following Falcon 9 breakup scenarios may be encountered:
- Off-nominal Scenario 1: Premature Falcon 9 failure results in an early abort, followed by aerodynamic breakup. Propellant is expected to be consumed similar to the nominal abort
- Off-nominal Scenario 2: Violation of autonomous flight termination criteria results in commanded destruct of Falcon 9, resulting in breakup of Falcon 9. Propellant is expected to be consumed similar to the nominal abort scenario. Dragon is anticipated to abort in this scenario. In general, failure cases are likely to result in an abort prior to an autonomous flight termination rule violation, as the abort triggers are more stringent by virtue of being designed to anticipate vehicle structural breakup from aerodynamic loads.
- Off-nominal Scenario 3: For early aborts where Falcon 9 velocity, and hence dynamic pressure, are still relatively low, if no autonomous flight termination rules are violated, Falcon 9 might impact the ocean’s surface intact. For aborts closer to the abort time frame, an intact impact is unlikely. In the event of Falcon 9 intact impact, propellant is expected to be consumed in the higher yield explosion resulting from propellant mixing upon impact.